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<!DOCTYPE html>
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<title>Chapter 36 Practical. Introduction to R | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</title>
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<meta name="twitter:title" content="Chapter 36 Practical. Introduction to R | Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi" />
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<meta name="author" content="Brad Duthie" />
<meta name="date" content="2024-02-24" />
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<ul class="summary">
<li><a href="./">Statistics with jamovi</a></li>
<li class="divider"></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preface</a>
<ul>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#structure"><i class="fa fa-check"></i>How this book is structured</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#datasets"><i class="fa fa-check"></i>Datasets used in this book</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#acknowledgements"><i class="fa fa-check"></i>Acknowledgements</a></li>
<li class="chapter" data-level="" data-path="index.html"><a href="index.html#author"><i class="fa fa-check"></i>About the author</a></li>
</ul></li>
<li class="part"><span><b>I Background mathematics and data organisation</b></span></li>
<li class="chapter" data-level="" data-path="Week1.html"><a href="Week1.html"><i class="fa fa-check"></i>Part I Overview</a></li>
<li class="chapter" data-level="1" data-path="Chapter_1.html"><a href="Chapter_1.html"><i class="fa fa-check"></i><b>1</b> Background mathematics</a>
<ul>
<li class="chapter" data-level="1.1" data-path="Chapter_1.html"><a href="Chapter_1.html#numbers-and-operations"><i class="fa fa-check"></i><b>1.1</b> Numbers and operations</a></li>
<li class="chapter" data-level="1.2" data-path="Chapter_1.html"><a href="Chapter_1.html#logarithms"><i class="fa fa-check"></i><b>1.2</b> Logarithms</a></li>
<li class="chapter" data-level="1.3" data-path="Chapter_1.html"><a href="Chapter_1.html#order-of-operations"><i class="fa fa-check"></i><b>1.3</b> Order of operations</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="Chapter_2.html"><a href="Chapter_2.html"><i class="fa fa-check"></i><b>2</b> Data organisation</a>
<ul>
<li class="chapter" data-level="2.1" data-path="Chapter_2.html"><a href="Chapter_2.html#tidy-data"><i class="fa fa-check"></i><b>2.1</b> Tidy data</a></li>
<li class="chapter" data-level="2.2" data-path="Chapter_2.html"><a href="Chapter_2.html#data-files"><i class="fa fa-check"></i><b>2.2</b> Data files</a></li>
<li class="chapter" data-level="2.3" data-path="Chapter_2.html"><a href="Chapter_2.html#managing-data-files"><i class="fa fa-check"></i><b>2.3</b> Managing data files</a></li>
</ul></li>
<li class="chapter" data-level="3" data-path="Chapter_3.html"><a href="Chapter_3.html"><i class="fa fa-check"></i><b>3</b> <em>Practical</em>. Preparing data</a>
<ul>
<li class="chapter" data-level="3.1" data-path="Chapter_3.html"><a href="Chapter_3.html#exercise-1-transferring-data-to-a-spreadsheet"><i class="fa fa-check"></i><b>3.1</b> Exercise 1: Transferring data to a spreadsheet</a></li>
<li class="chapter" data-level="3.2" data-path="Chapter_3.html"><a href="Chapter_3.html#exercise-2-making-spreadsheet-data-tidy"><i class="fa fa-check"></i><b>3.2</b> Exercise 2: Making spreadsheet data tidy</a></li>
<li class="chapter" data-level="3.3" data-path="Chapter_3.html"><a href="Chapter_3.html#exercise-3-making-data-tidy-again"><i class="fa fa-check"></i><b>3.3</b> Exercise 3: Making data tidy again</a></li>
<li class="chapter" data-level="3.4" data-path="Chapter_3.html"><a href="Chapter_3.html#exercise-4-tidy-data-and-spreadsheet-calculations"><i class="fa fa-check"></i><b>3.4</b> Exercise 4: Tidy data and spreadsheet calculations</a></li>
<li class="chapter" data-level="3.5" data-path="Chapter_3.html"><a href="Chapter_3.html#summary"><i class="fa fa-check"></i><b>3.5</b> Summary</a></li>
</ul></li>
<li class="part"><span><b>II Statistical concepts</b></span></li>
<li class="chapter" data-level="" data-path="Week2.html"><a href="Week2.html"><i class="fa fa-check"></i>Part II Overview</a></li>
<li class="chapter" data-level="4" data-path="Chapter_4.html"><a href="Chapter_4.html"><i class="fa fa-check"></i><b>4</b> Populations and samples</a></li>
<li class="chapter" data-level="5" data-path="Chapter_5.html"><a href="Chapter_5.html"><i class="fa fa-check"></i><b>5</b> Types of variables</a></li>
<li class="chapter" data-level="6" data-path="Chapter_6.html"><a href="Chapter_6.html"><i class="fa fa-check"></i><b>6</b> Accuracy, precision, and units</a>
<ul>
<li class="chapter" data-level="6.1" data-path="Chapter_6.html"><a href="Chapter_6.html#accuracy"><i class="fa fa-check"></i><b>6.1</b> Accuracy</a></li>
<li class="chapter" data-level="6.2" data-path="Chapter_6.html"><a href="Chapter_6.html#precision"><i class="fa fa-check"></i><b>6.2</b> Precision</a></li>
<li class="chapter" data-level="6.3" data-path="Chapter_6.html"><a href="Chapter_6.html#systems-of-units"><i class="fa fa-check"></i><b>6.3</b> Systems of units</a></li>
<li class="chapter" data-level="6.4" data-path="Chapter_6.html"><a href="Chapter_6.html#other-examples-of-units"><i class="fa fa-check"></i><b>6.4</b> Other examples of units</a>
<ul>
<li class="chapter" data-level="6.4.1" data-path="Chapter_6.html"><a href="Chapter_6.html#units-of-density"><i class="fa fa-check"></i><b>6.4.1</b> Units of density</a></li>
<li class="chapter" data-level="6.4.2" data-path="Chapter_6.html"><a href="Chapter_6.html#mass-of-metal-discharged-from-a-catchment"><i class="fa fa-check"></i><b>6.4.2</b> Mass of metal discharged from a catchment</a></li>
<li class="chapter" data-level="6.4.3" data-path="Chapter_6.html"><a href="Chapter_6.html#soil-carbon-inventories"><i class="fa fa-check"></i><b>6.4.3</b> Soil carbon inventories</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="7" data-path="Chapter_7.html"><a href="Chapter_7.html"><i class="fa fa-check"></i><b>7</b> Uncertainty propagation</a>
<ul>
<li class="chapter" data-level="7.1" data-path="Chapter_7.html"><a href="Chapter_7.html#adding-or-subtracting-errors"><i class="fa fa-check"></i><b>7.1</b> Adding or subtracting errors</a></li>
<li class="chapter" data-level="7.2" data-path="Chapter_7.html"><a href="Chapter_7.html#multiplying-or-dividing-errors"><i class="fa fa-check"></i><b>7.2</b> Multiplying or dividing errors</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="Chapter_8.html"><a href="Chapter_8.html"><i class="fa fa-check"></i><b>8</b> <em>Practical</em>. Introduction to jamovi</a>
<ul>
<li class="chapter" data-level="8.1" data-path="Chapter_8.html"><a href="Chapter_8.html#summary_statistics_02"><i class="fa fa-check"></i><b>8.1</b> Exercise for summary statistics</a></li>
<li class="chapter" data-level="8.2" data-path="Chapter_8.html"><a href="Chapter_8.html#transforming_variables_02"><i class="fa fa-check"></i><b>8.2</b> Exercise on transforming variables</a></li>
<li class="chapter" data-level="8.3" data-path="Chapter_8.html"><a href="Chapter_8.html#computing_variables_02"><i class="fa fa-check"></i><b>8.3</b> Exercise on computing variables</a></li>
<li class="chapter" data-level="8.4" data-path="Chapter_8.html"><a href="Chapter_8.html#summary-1"><i class="fa fa-check"></i><b>8.4</b> Summary</a></li>
</ul></li>
<li class="part"><span><b>III Summary statistics</b></span></li>
<li class="chapter" data-level="" data-path="Week3.html"><a href="Week3.html"><i class="fa fa-check"></i>Part III Overview</a></li>
<li class="chapter" data-level="9" data-path="Chapter_9.html"><a href="Chapter_9.html"><i class="fa fa-check"></i><b>9</b> Decimal places, significant figures, and rounding</a>
<ul>
<li class="chapter" data-level="9.1" data-path="Chapter_9.html"><a href="Chapter_9.html#decimal-places-and-significant-figures"><i class="fa fa-check"></i><b>9.1</b> Decimal places and significant figures</a></li>
<li class="chapter" data-level="9.2" data-path="Chapter_9.html"><a href="Chapter_9.html#rounding"><i class="fa fa-check"></i><b>9.2</b> Rounding</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="Chapter_10.html"><a href="Chapter_10.html"><i class="fa fa-check"></i><b>10</b> Graphs</a>
<ul>
<li class="chapter" data-level="10.1" data-path="Chapter_10.html"><a href="Chapter_10.html#histograms"><i class="fa fa-check"></i><b>10.1</b> Histograms</a></li>
<li class="chapter" data-level="10.2" data-path="Chapter_10.html"><a href="Chapter_10.html#barplots-and-pie-charts"><i class="fa fa-check"></i><b>10.2</b> Barplots and pie charts</a></li>
<li class="chapter" data-level="10.3" data-path="Chapter_10.html"><a href="Chapter_10.html#box-whisker-plots"><i class="fa fa-check"></i><b>10.3</b> Box-whisker plots</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="Chapter_11.html"><a href="Chapter_11.html"><i class="fa fa-check"></i><b>11</b> Measures of central tendency</a>
<ul>
<li class="chapter" data-level="11.1" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mean"><i class="fa fa-check"></i><b>11.1</b> The mean</a></li>
<li class="chapter" data-level="11.2" data-path="Chapter_11.html"><a href="Chapter_11.html#the-mode"><i class="fa fa-check"></i><b>11.2</b> The mode</a></li>
<li class="chapter" data-level="11.3" data-path="Chapter_11.html"><a href="Chapter_11.html#the-median-and-quantiles"><i class="fa fa-check"></i><b>11.3</b> The median and quantiles</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="Chapter_12.html"><a href="Chapter_12.html"><i class="fa fa-check"></i><b>12</b> Measures of spread</a>
<ul>
<li class="chapter" data-level="12.1" data-path="Chapter_12.html"><a href="Chapter_12.html#the-range"><i class="fa fa-check"></i><b>12.1</b> The range</a></li>
<li class="chapter" data-level="12.2" data-path="Chapter_12.html"><a href="Chapter_12.html#the-inter-quartile-range"><i class="fa fa-check"></i><b>12.2</b> The inter-quartile range</a></li>
<li class="chapter" data-level="12.3" data-path="Chapter_12.html"><a href="Chapter_12.html#the-variance"><i class="fa fa-check"></i><b>12.3</b> The variance</a></li>
<li class="chapter" data-level="12.4" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-deviation"><i class="fa fa-check"></i><b>12.4</b> The standard deviation</a></li>
<li class="chapter" data-level="12.5" data-path="Chapter_12.html"><a href="Chapter_12.html#the-coefficient-of-variation"><i class="fa fa-check"></i><b>12.5</b> The coefficient of variation</a></li>
<li class="chapter" data-level="12.6" data-path="Chapter_12.html"><a href="Chapter_12.html#the-standard-error"><i class="fa fa-check"></i><b>12.6</b> The standard error</a></li>
</ul></li>
<li class="chapter" data-level="13" data-path="skew-and-kurtosis.html"><a href="skew-and-kurtosis.html"><i class="fa fa-check"></i><b>13</b> Skew and Kurtosis</a>
<ul>
<li class="chapter" data-level="13.1" data-path="skew-and-kurtosis.html"><a href="skew-and-kurtosis.html#skew"><i class="fa fa-check"></i><b>13.1</b> Skew</a></li>
<li class="chapter" data-level="13.2" data-path="skew-and-kurtosis.html"><a href="skew-and-kurtosis.html#kurtosis"><i class="fa fa-check"></i><b>13.2</b> Kurtosis</a></li>
<li class="chapter" data-level="13.3" data-path="skew-and-kurtosis.html"><a href="skew-and-kurtosis.html#moments"><i class="fa fa-check"></i><b>13.3</b> Moments</a></li>
</ul></li>
<li class="chapter" data-level="14" data-path="Chapter_14.html"><a href="Chapter_14.html"><i class="fa fa-check"></i><b>14</b> <em>Practical</em>. Plotting and statistical summaries in jamovi</a>
<ul>
<li class="chapter" data-level="14.1" data-path="Chapter_14.html"><a href="Chapter_14.html#reorganise-the-dataset-into-a-tidy-format"><i class="fa fa-check"></i><b>14.1</b> Reorganise the dataset into a tidy format</a></li>
<li class="chapter" data-level="14.2" data-path="Chapter_14.html"><a href="Chapter_14.html#histograms-and-box-whisker-plots"><i class="fa fa-check"></i><b>14.2</b> Histograms and box-whisker plots</a></li>
<li class="chapter" data-level="14.3" data-path="Chapter_14.html"><a href="Chapter_14.html#calculate-summary-statistics"><i class="fa fa-check"></i><b>14.3</b> Calculate summary statistics</a></li>
<li class="chapter" data-level="14.4" data-path="Chapter_14.html"><a href="Chapter_14.html#reporting-decimals-and-significant-figures"><i class="fa fa-check"></i><b>14.4</b> Reporting decimals and significant figures</a></li>
<li class="chapter" data-level="14.5" data-path="Chapter_14.html"><a href="Chapter_14.html#comparing-across-sites"><i class="fa fa-check"></i><b>14.5</b> Comparing across sites</a></li>
</ul></li>
<li class="part"><span><b>IV Probability models and the Central Limit Theorem</b></span></li>
<li class="chapter" data-level="" data-path="Week4.html"><a href="Week4.html"><i class="fa fa-check"></i>Part IV Overview</a></li>
<li class="chapter" data-level="15" data-path="Chapter_15.html"><a href="Chapter_15.html"><i class="fa fa-check"></i><b>15</b> Introduction to probability models</a>
<ul>
<li class="chapter" data-level="15.1" data-path="Chapter_15.html"><a href="Chapter_15.html#an-instructive-example"><i class="fa fa-check"></i><b>15.1</b> An instructive example</a></li>
<li class="chapter" data-level="15.2" data-path="Chapter_15.html"><a href="Chapter_15.html#biological-applications"><i class="fa fa-check"></i><b>15.2</b> Biological applications</a></li>
<li class="chapter" data-level="15.3" data-path="Chapter_15.html"><a href="Chapter_15.html#sampling-with-and-without-replacement"><i class="fa fa-check"></i><b>15.3</b> Sampling with and without replacement</a></li>
<li class="chapter" data-level="15.4" data-path="Chapter_15.html"><a href="Chapter_15.html#probability-distributions"><i class="fa fa-check"></i><b>15.4</b> Probability distributions</a>
<ul>
<li class="chapter" data-level="15.4.1" data-path="Chapter_15.html"><a href="Chapter_15.html#binomial-distribution"><i class="fa fa-check"></i><b>15.4.1</b> Binomial distribution</a></li>
<li class="chapter" data-level="15.4.2" data-path="Chapter_15.html"><a href="Chapter_15.html#poisson-distribution"><i class="fa fa-check"></i><b>15.4.2</b> Poisson distribution</a></li>
<li class="chapter" data-level="15.4.3" data-path="Chapter_15.html"><a href="Chapter_15.html#uniform-distribution"><i class="fa fa-check"></i><b>15.4.3</b> Uniform distribution</a></li>
<li class="chapter" data-level="15.4.4" data-path="Chapter_15.html"><a href="Chapter_15.html#normal-distribution"><i class="fa fa-check"></i><b>15.4.4</b> Normal distribution</a></li>
</ul></li>
<li class="chapter" data-level="15.5" data-path="Chapter_15.html"><a href="Chapter_15.html#summary-2"><i class="fa fa-check"></i><b>15.5</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="16" data-path="Chapter_16.html"><a href="Chapter_16.html"><i class="fa fa-check"></i><b>16</b> The Central Limit Theorem (CLT)</a>
<ul>
<li class="chapter" data-level="16.1" data-path="Chapter_16.html"><a href="Chapter_16.html#the-distribution-of-means-is-normal"><i class="fa fa-check"></i><b>16.1</b> The distribution of means is normal</a></li>
<li class="chapter" data-level="16.2" data-path="Chapter_16.html"><a href="Chapter_16.html#probability-and-z-scores"><i class="fa fa-check"></i><b>16.2</b> Probability and z-scores</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="Chapter_17.html"><a href="Chapter_17.html"><i class="fa fa-check"></i><b>17</b> <em>Practical</em>. Probability and simulation</a>
<ul>
<li class="chapter" data-level="17.1" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-dataset"><i class="fa fa-check"></i><b>17.1</b> Probabilities from a dataset</a></li>
<li class="chapter" data-level="17.2" data-path="Chapter_17.html"><a href="Chapter_17.html#probabilities-from-a-normal-distribution"><i class="fa fa-check"></i><b>17.2</b> Probabilities from a normal distribution</a></li>
<li class="chapter" data-level="17.3" data-path="Chapter_17.html"><a href="Chapter_17.html#central-limit-theorem"><i class="fa fa-check"></i><b>17.3</b> Central limit theorem</a></li>
</ul></li>
<li class="part"><span><b>V Statistical inference</b></span></li>
<li class="chapter" data-level="" data-path="Week5.html"><a href="Week5.html"><i class="fa fa-check"></i>Part V Overview</a></li>
<li class="chapter" data-level="18" data-path="Chapter_18.html"><a href="Chapter_18.html"><i class="fa fa-check"></i><b>18</b> Confidence intervals (CIs)</a>
<ul>
<li class="chapter" data-level="18.1" data-path="Chapter_18.html"><a href="Chapter_18.html#normal-distribution-cis"><i class="fa fa-check"></i><b>18.1</b> Normal distribution CIs</a></li>
<li class="chapter" data-level="18.2" data-path="Chapter_18.html"><a href="Chapter_18.html#binomial-distribution-cis"><i class="fa fa-check"></i><b>18.2</b> Binomial distribution CIs</a></li>
</ul></li>
<li class="chapter" data-level="19" data-path="Chapter_19.html"><a href="Chapter_19.html"><i class="fa fa-check"></i><b>19</b> The t-interval</a></li>
<li class="chapter" data-level="20" data-path="Chapter_20.html"><a href="Chapter_20.html"><i class="fa fa-check"></i><b>20</b> <em>Practical</em>. z- and t- intervals</a>
<ul>
<li class="chapter" data-level="20.1" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-with-distraction"><i class="fa fa-check"></i><b>20.1</b> Confidence intervals with distrACTION</a></li>
<li class="chapter" data-level="20.2" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-from-z--and-t-scores"><i class="fa fa-check"></i><b>20.2</b> Confidence intervals from z- and t-scores</a></li>
<li class="chapter" data-level="20.3" data-path="Chapter_20.html"><a href="Chapter_20.html#confidence-intervals-for-different-sample-sizes-t--and-z-"><i class="fa fa-check"></i><b>20.3</b> Confidence intervals for different sample sizes (t- and z-)</a></li>
<li class="chapter" data-level="20.4" data-path="Chapter_20.html"><a href="Chapter_20.html#proportion-confidence-intervals"><i class="fa fa-check"></i><b>20.4</b> Proportion confidence intervals</a></li>
<li class="chapter" data-level="20.5" data-path="Chapter_20.html"><a href="Chapter_20.html#another-proportion-confidence-interval"><i class="fa fa-check"></i><b>20.5</b> Another proportion confidence interval</a></li>
</ul></li>
<li class="part"><span><b>VI Hypothesis testing</b></span></li>
<li class="chapter" data-level="" data-path="Week6.html"><a href="Week6.html"><i class="fa fa-check"></i>Part VI Overview</a></li>
<li class="chapter" data-level="21" data-path="Chapter_21.html"><a href="Chapter_21.html"><i class="fa fa-check"></i><b>21</b> What is hypothesis testing?</a>
<ul>
<li class="chapter" data-level="21.1" data-path="Chapter_21.html"><a href="Chapter_21.html#how-ridiculous-is-our-hypothesis"><i class="fa fa-check"></i><b>21.1</b> How ridiculous is our hypothesis?</a></li>
<li class="chapter" data-level="21.2" data-path="Chapter_21.html"><a href="Chapter_21.html#statistical-hypothesis-testing"><i class="fa fa-check"></i><b>21.2</b> Statistical hypothesis testing</a></li>
<li class="chapter" data-level="21.3" data-path="Chapter_21.html"><a href="Chapter_21.html#p-values-false-positives-and-power"><i class="fa fa-check"></i><b>21.3</b> P-values, false positives, and power</a></li>
</ul></li>
<li class="chapter" data-level="22" data-path="Chapter_22.html"><a href="Chapter_22.html"><i class="fa fa-check"></i><b>22</b> The t-test</a>
<ul>
<li class="chapter" data-level="22.1" data-path="Chapter_22.html"><a href="Chapter_22.html#one-sample-t-test"><i class="fa fa-check"></i><b>22.1</b> One sample t-test</a></li>
<li class="chapter" data-level="22.2" data-path="Chapter_22.html"><a href="Chapter_22.html#independent-samples-t-test"><i class="fa fa-check"></i><b>22.2</b> Independent samples t-test</a></li>
<li class="chapter" data-level="22.3" data-path="Chapter_22.html"><a href="Chapter_22.html#paired-sample-t-test"><i class="fa fa-check"></i><b>22.3</b> Paired sample t-test</a></li>
<li class="chapter" data-level="22.4" data-path="Chapter_22.html"><a href="Chapter_22.html#assumptions-of-t-tests"><i class="fa fa-check"></i><b>22.4</b> Assumptions of t-tests</a></li>
<li class="chapter" data-level="22.5" data-path="Chapter_22.html"><a href="Chapter_22.html#non-parametric-alternatives"><i class="fa fa-check"></i><b>22.5</b> Non-parametric alternatives</a>
<ul>
<li class="chapter" data-level="22.5.1" data-path="Chapter_22.html"><a href="Chapter_22.html#wilcoxon-test"><i class="fa fa-check"></i><b>22.5.1</b> Wilcoxon test</a></li>
<li class="chapter" data-level="22.5.2" data-path="Chapter_22.html"><a href="Chapter_22.html#mann-whitney-u-test"><i class="fa fa-check"></i><b>22.5.2</b> Mann-Whitney U test</a></li>
</ul></li>
<li class="chapter" data-level="22.6" data-path="Chapter_22.html"><a href="Chapter_22.html#summary-3"><i class="fa fa-check"></i><b>22.6</b> Summary</a></li>
</ul></li>
<li class="chapter" data-level="23" data-path="Chapter_23.html"><a href="Chapter_23.html"><i class="fa fa-check"></i><b>23</b> <em>Practical</em>. Hypothesis testing and t-tests</a>
<ul>
<li class="chapter" data-level="23.1" data-path="Chapter_23.html"><a href="Chapter_23.html#exercise-on-a-simple-one-sample-t-test"><i class="fa fa-check"></i><b>23.1</b> Exercise on a simple one sample t-test</a></li>
<li class="chapter" data-level="23.2" data-path="Chapter_23.html"><a href="Chapter_23.html#exercise-on-a-paired-t-test"><i class="fa fa-check"></i><b>23.2</b> Exercise on a paired t-test</a></li>
<li class="chapter" data-level="23.3" data-path="Chapter_23.html"><a href="Chapter_23.html#wilcoxon-test-1"><i class="fa fa-check"></i><b>23.3</b> Wilcoxon test</a></li>
<li class="chapter" data-level="23.4" data-path="Chapter_23.html"><a href="Chapter_23.html#independent-samples-t-test-1"><i class="fa fa-check"></i><b>23.4</b> Independent samples t-test</a></li>
<li class="chapter" data-level="23.5" data-path="Chapter_23.html"><a href="Chapter_23.html#mann-whitney-u-test-1"><i class="fa fa-check"></i><b>23.5</b> Mann-Whitney U Test</a></li>
</ul></li>
<li class="part"><span><b>VII Analysis of Variance (ANOVA)</b></span></li>
<li class="chapter" data-level="" data-path="Week8.html"><a href="Week8.html"><i class="fa fa-check"></i>Part VII Overview</a></li>
<li class="chapter" data-level="24" data-path="Chapter_24.html"><a href="Chapter_24.html"><i class="fa fa-check"></i><b>24</b> Analysis of variance</a>
<ul>
<li class="chapter" data-level="24.1" data-path="Chapter_24.html"><a href="Chapter_24.html#the-f-distribution"><i class="fa fa-check"></i><b>24.1</b> The F-distribution</a></li>
<li class="chapter" data-level="24.2" data-path="Chapter_24.html"><a href="Chapter_24.html#one-way-anova"><i class="fa fa-check"></i><b>24.2</b> One-way ANOVA</a>
<ul>
<li class="chapter" data-level="24.2.1" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-among-groups"><i class="fa fa-check"></i><b>24.2.1</b> ANOVA mean variance among groups</a></li>
<li class="chapter" data-level="24.2.2" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-mean-variance-within-groups"><i class="fa fa-check"></i><b>24.2.2</b> ANOVA mean variance within groups</a></li>
<li class="chapter" data-level="24.2.3" data-path="Chapter_24.html"><a href="Chapter_24.html#anova-f-statistic-calculation"><i class="fa fa-check"></i><b>24.2.3</b> ANOVA F statistic calculation</a></li>
</ul></li>
<li class="chapter" data-level="24.3" data-path="Chapter_24.html"><a href="Chapter_24.html#assumptions-of-anova"><i class="fa fa-check"></i><b>24.3</b> Assumptions of ANOVA</a></li>
</ul></li>
<li class="chapter" data-level="25" data-path="Chapter_25.html"><a href="Chapter_25.html"><i class="fa fa-check"></i><b>25</b> Multiple comparisons</a></li>
<li class="chapter" data-level="26" data-path="Chapter_26.html"><a href="Chapter_26.html"><i class="fa fa-check"></i><b>26</b> Kruskall-Wallis H test</a></li>
<li class="chapter" data-level="27" data-path="Chapter_27.html"><a href="Chapter_27.html"><i class="fa fa-check"></i><b>27</b> Two-way ANOVA</a></li>
<li class="chapter" data-level="28" data-path="Chapter_28.html"><a href="Chapter_28.html"><i class="fa fa-check"></i><b>28</b> <em>Practical</em>. ANOVA and associated tests</a>
<ul>
<li class="chapter" data-level="28.1" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-site"><i class="fa fa-check"></i><b>28.1</b> One-way ANOVA (site)</a></li>
<li class="chapter" data-level="28.2" data-path="Chapter_28.html"><a href="Chapter_28.html#one-way-anova-profile"><i class="fa fa-check"></i><b>28.2</b> One-way ANOVA (profile)</a></li>
<li class="chapter" data-level="28.3" data-path="Chapter_28.html"><a href="Chapter_28.html#multiple-comparisons"><i class="fa fa-check"></i><b>28.3</b> Multiple comparisons</a></li>
<li class="chapter" data-level="28.4" data-path="Chapter_28.html"><a href="Chapter_28.html#kruskall-wallis-h-test"><i class="fa fa-check"></i><b>28.4</b> Kruskall-Wallis H test</a></li>
<li class="chapter" data-level="28.5" data-path="Chapter_28.html"><a href="Chapter_28.html#two-way-anova"><i class="fa fa-check"></i><b>28.5</b> Two-way ANOVA</a></li>
</ul></li>
<li class="part"><span><b>VIII Counts and Correlation</b></span></li>
<li class="chapter" data-level="" data-path="Week9.html"><a href="Week9.html"><i class="fa fa-check"></i>Part VIII Overview</a></li>
<li class="chapter" data-level="29" data-path="Chapter_29.html"><a href="Chapter_29.html"><i class="fa fa-check"></i><b>29</b> Frequency and count data</a>
<ul>
<li class="chapter" data-level="29.1" data-path="Chapter_29.html"><a href="Chapter_29.html#the-chi-square-distribution"><i class="fa fa-check"></i><b>29.1</b> The Chi-square distribution</a></li>
<li class="chapter" data-level="29.2" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-squared-goodness-of-fit"><i class="fa fa-check"></i><b>29.2</b> Chi-squared goodness of fit</a></li>
<li class="chapter" data-level="29.3" data-path="Chapter_29.html"><a href="Chapter_29.html#chi-squared-test-of-association"><i class="fa fa-check"></i><b>29.3</b> Chi-squared test of association</a></li>
</ul></li>
<li class="chapter" data-level="30" data-path="Chapter_30.html"><a href="Chapter_30.html"><i class="fa fa-check"></i><b>30</b> Correlation</a>
<ul>
<li class="chapter" data-level="30.1" data-path="Chapter_30.html"><a href="Chapter_30.html#scatterplots"><i class="fa fa-check"></i><b>30.1</b> Scatterplots</a></li>
<li class="chapter" data-level="30.2" data-path="Chapter_30.html"><a href="Chapter_30.html#the-correlation-coefficient"><i class="fa fa-check"></i><b>30.2</b> The correlation coefficient</a>
<ul>
<li class="chapter" data-level="30.2.1" data-path="Chapter_30.html"><a href="Chapter_30.html#pearson-product-moment-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.1</b> Pearson product moment correlation coefficient</a></li>
<li class="chapter" data-level="30.2.2" data-path="Chapter_30.html"><a href="Chapter_30.html#spearman-rank-correlation-coefficient"><i class="fa fa-check"></i><b>30.2.2</b> Spearman rank correlation coefficient</a></li>
</ul></li>
<li class="chapter" data-level="30.3" data-path="Chapter_30.html"><a href="Chapter_30.html#correlation-hypothesis-testing"><i class="fa fa-check"></i><b>30.3</b> Correlation hypothesis testing</a></li>
</ul></li>
<li class="chapter" data-level="31" data-path="Chapter_31.html"><a href="Chapter_31.html"><i class="fa fa-check"></i><b>31</b> <em>Practical</em>. Analysis of counts and correlations</a>
<ul>
<li class="chapter" data-level="31.1" data-path="Chapter_31.html"><a href="Chapter_31.html#survival-goodness-of-fit"><i class="fa fa-check"></i><b>31.1</b> Survival goodness of fit</a></li>
<li class="chapter" data-level="31.2" data-path="Chapter_31.html"><a href="Chapter_31.html#colony-goodness-of-fit"><i class="fa fa-check"></i><b>31.2</b> Colony goodness of fit</a></li>
<li class="chapter" data-level="31.3" data-path="Chapter_31.html"><a href="Chapter_31.html#chi-square-test-of-association"><i class="fa fa-check"></i><b>31.3</b> Chi-Square test of association</a></li>
<li class="chapter" data-level="31.4" data-path="Chapter_31.html"><a href="Chapter_31.html#pearson-product-moment-correlation-test"><i class="fa fa-check"></i><b>31.4</b> Pearson product moment correlation test</a></li>
<li class="chapter" data-level="31.5" data-path="Chapter_31.html"><a href="Chapter_31.html#spearman-rank-correlation-test"><i class="fa fa-check"></i><b>31.5</b> Spearman rank correlation test</a></li>
<li class="chapter" data-level="31.6" data-path="Chapter_31.html"><a href="Chapter_31.html#untidy-goodness-of-fit"><i class="fa fa-check"></i><b>31.6</b> Untidy goodness of fit</a></li>
</ul></li>
<li class="part"><span><b>IX Linear Regression</b></span></li>
<li class="chapter" data-level="" data-path="Week10.html"><a href="Week10.html"><i class="fa fa-check"></i>Part IX Overview</a></li>
<li class="chapter" data-level="32" data-path="Chapter_32.html"><a href="Chapter_32.html"><i class="fa fa-check"></i><b>32</b> Simple linear regression</a>
<ul>
<li class="chapter" data-level="32.1" data-path="Chapter_32.html"><a href="Chapter_32.html#visual-interpretation-of-regression"><i class="fa fa-check"></i><b>32.1</b> Visual interpretation of regression</a></li>
<li class="chapter" data-level="32.2" data-path="Chapter_32.html"><a href="Chapter_32.html#intercepts-slopes-and-residuals"><i class="fa fa-check"></i><b>32.2</b> Intercepts, slopes, and residuals</a></li>
<li class="chapter" data-level="32.3" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-coefficients"><i class="fa fa-check"></i><b>32.3</b> Regression coefficients</a></li>
<li class="chapter" data-level="32.4" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-line-calculation"><i class="fa fa-check"></i><b>32.4</b> Regression line calculation</a></li>
<li class="chapter" data-level="32.5" data-path="Chapter_32.html"><a href="Chapter_32.html#coefficient-of-determination"><i class="fa fa-check"></i><b>32.5</b> Coefficient of determination</a></li>
<li class="chapter" data-level="32.6" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-assumptions"><i class="fa fa-check"></i><b>32.6</b> Regression assumptions</a></li>
<li class="chapter" data-level="32.7" data-path="Chapter_32.html"><a href="Chapter_32.html#regression-hypothesis-testing"><i class="fa fa-check"></i><b>32.7</b> Regression hypothesis testing</a>
<ul>
<li class="chapter" data-level="32.7.1" data-path="Chapter_32.html"><a href="Chapter_32.html#overall-model-significance"><i class="fa fa-check"></i><b>32.7.1</b> Overall model significance</a></li>
<li class="chapter" data-level="32.7.2" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-intercept"><i class="fa fa-check"></i><b>32.7.2</b> Significance of the intercept</a></li>
<li class="chapter" data-level="32.7.3" data-path="Chapter_32.html"><a href="Chapter_32.html#significance-of-the-slope"><i class="fa fa-check"></i><b>32.7.3</b> Significance of the slope</a></li>
<li class="chapter" data-level="32.7.4" data-path="Chapter_32.html"><a href="Chapter_32.html#simple-regression-output"><i class="fa fa-check"></i><b>32.7.4</b> Simple regression output</a></li>
</ul></li>
<li class="chapter" data-level="32.8" data-path="Chapter_32.html"><a href="Chapter_32.html#prediction-with-linear-models"><i class="fa fa-check"></i><b>32.8</b> Prediction with linear models</a></li>
<li class="chapter" data-level="32.9" data-path="Chapter_32.html"><a href="Chapter_32.html#conclusion"><i class="fa fa-check"></i><b>32.9</b> Conclusion</a></li>
</ul></li>
<li class="chapter" data-level="33" data-path="Chapter_33.html"><a href="Chapter_33.html"><i class="fa fa-check"></i><b>33</b> Multiple regression</a>
<ul>
<li class="chapter" data-level="33.1" data-path="Chapter_33.html"><a href="Chapter_33.html#adjusted-coefficient-of-determination"><i class="fa fa-check"></i><b>33.1</b> Adjusted coefficient of determination</a></li>
</ul></li>
<li class="chapter" data-level="34" data-path="Chapter_34.html"><a href="Chapter_34.html"><i class="fa fa-check"></i><b>34</b> <em>Practical</em>. Using regression</a>
<ul>
<li class="chapter" data-level="34.1" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-soil-depth"><i class="fa fa-check"></i><b>34.1</b> Predicting pyrogenic carbon from soil depth</a></li>
<li class="chapter" data-level="34.2" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-pyrogenic-carbon-from-fire-frequency"><i class="fa fa-check"></i><b>34.2</b> Predicting pyrogenic carbon from fire frequency</a></li>
<li class="chapter" data-level="34.3" data-path="Chapter_34.html"><a href="Chapter_34.html#multiple-regression-depth-and-fire-frequency"><i class="fa fa-check"></i><b>34.3</b> Multiple regression depth and fire frequency</a></li>
<li class="chapter" data-level="34.4" data-path="Chapter_34.html"><a href="Chapter_34.html#large-multiple-regression"><i class="fa fa-check"></i><b>34.4</b> Large multiple regression</a></li>
<li class="chapter" data-level="34.5" data-path="Chapter_34.html"><a href="Chapter_34.html#predicting-temperature-from-fire-frequency"><i class="fa fa-check"></i><b>34.5</b> Predicting temperature from fire frequency</a></li>
</ul></li>
<li class="part"><span><b>X Randomisation approaches</b></span></li>
<li class="chapter" data-level="" data-path="Week11.html"><a href="Week11.html"><i class="fa fa-check"></i>Part X Overview</a></li>
<li class="chapter" data-level="35" data-path="Chapter_35.html"><a href="Chapter_35.html"><i class="fa fa-check"></i><b>35</b> Randomisation</a>
<ul>
<li class="chapter" data-level="35.1" data-path="Chapter_35.html"><a href="Chapter_35.html#summary-of-parametric-hypothesis-testing"><i class="fa fa-check"></i><b>35.1</b> Summary of parametric hypothesis testing</a></li>
<li class="chapter" data-level="35.2" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-approach"><i class="fa fa-check"></i><b>35.2</b> Randomisation approach</a></li>
<li class="chapter" data-level="35.3" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-for-hypothesis-testing"><i class="fa fa-check"></i><b>35.3</b> Randomisation for hypothesis testing</a></li>
<li class="chapter" data-level="35.4" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-assumptions"><i class="fa fa-check"></i><b>35.4</b> Randomisation assumptions</a></li>
<li class="chapter" data-level="35.5" data-path="Chapter_35.html"><a href="Chapter_35.html#bootstrapping"><i class="fa fa-check"></i><b>35.5</b> Bootstrapping</a></li>
<li class="chapter" data-level="35.6" data-path="Chapter_35.html"><a href="Chapter_35.html#monte-carlo"><i class="fa fa-check"></i><b>35.6</b> Monte Carlo</a></li>
<li class="chapter" data-level="35.7" data-path="Chapter_35.html"><a href="Chapter_35.html#randomisation-conclusions"><i class="fa fa-check"></i><b>35.7</b> Randomisation conclusions</a></li>
</ul></li>
<li class="chapter" data-level="36" data-path="Chapter_36.html"><a href="Chapter_36.html"><i class="fa fa-check"></i><b>36</b> <em>Practical</em>. Introduction to R</a>
<ul>
<li class="chapter" data-level="36.1" data-path="Chapter_36.html"><a href="Chapter_36.html#getting-used-to-the-r-interface"><i class="fa fa-check"></i><b>36.1</b> Getting used to the R interface</a></li>
<li class="chapter" data-level="36.2" data-path="Chapter_36.html"><a href="Chapter_36.html#assigning-variables-in-the-r-console"><i class="fa fa-check"></i><b>36.2</b> Assigning variables in the R console</a></li>
<li class="chapter" data-level="36.3" data-path="Chapter_36.html"><a href="Chapter_36.html#some-descriptive-statistics"><i class="fa fa-check"></i><b>36.3</b> Some descriptive statistics</a></li>
<li class="chapter" data-level="36.4" data-path="Chapter_36.html"><a href="Chapter_36.html#bootstrapping-confidence-intervals"><i class="fa fa-check"></i><b>36.4</b> Bootstrapping confidence intervals</a></li>
</ul></li>
<li class="part"><span><b>XI Experimental Design and Statistical Reporting</b></span></li>
<li class="chapter" data-level="" data-path="Week12.html"><a href="Week12.html"><i class="fa fa-check"></i>Part XI Overview</a></li>
<li class="chapter" data-level="37" data-path="Chapter_37.html"><a href="Chapter_37.html"><i class="fa fa-check"></i><b>37</b> Experimental design</a>
<ul>
<li class="chapter" data-level="37.1" data-path="Chapter_37.html"><a href="Chapter_37.html#before-collecting-data"><i class="fa fa-check"></i><b>37.1</b> Before collecting data</a></li>
</ul></li>
<li class="chapter" data-level="38" data-path="Chapter_38.html"><a href="Chapter_38.html"><i class="fa fa-check"></i><b>38</b> Reporting statistics</a>
<ul>
<li class="chapter" data-level="38.1" data-path="Chapter_38.html"><a href="Chapter_38.html#statistical-reporting"><i class="fa fa-check"></i><b>38.1</b> Statistical reporting</a>
<ul>
<li class="chapter" data-level="38.1.1" data-path="Chapter_38.html"><a href="Chapter_38.html#abstract"><i class="fa fa-check"></i><b>38.1.1</b> Abstract</a></li>
<li class="chapter" data-level="38.1.2" data-path="Chapter_38.html"><a href="Chapter_38.html#introduction"><i class="fa fa-check"></i><b>38.1.2</b> Introduction</a></li>
<li class="chapter" data-level="38.1.3" data-path="Chapter_38.html"><a href="Chapter_38.html#methods"><i class="fa fa-check"></i><b>38.1.3</b> Methods</a></li>
<li class="chapter" data-level="38.1.4" data-path="Chapter_38.html"><a href="Chapter_38.html#results"><i class="fa fa-check"></i><b>38.1.4</b> Results</a></li>
<li class="chapter" data-level="38.1.5" data-path="Chapter_38.html"><a href="Chapter_38.html#discussion"><i class="fa fa-check"></i><b>38.1.5</b> Discussion</a></li>
</ul></li>
<li class="chapter" data-level="38.2" data-path="Chapter_38.html"><a href="Chapter_38.html#figures-and-tables"><i class="fa fa-check"></i><b>38.2</b> Figures and tables</a>
<ul>
<li class="chapter" data-level="38.2.1" data-path="Chapter_38.html"><a href="Chapter_38.html#figures"><i class="fa fa-check"></i><b>38.2.1</b> Figures</a></li>
<li class="chapter" data-level="38.2.2" data-path="Chapter_38.html"><a href="Chapter_38.html#tables"><i class="fa fa-check"></i><b>38.2.2</b> Tables</a></li>
</ul></li>
<li class="chapter" data-level="38.3" data-path="Chapter_38.html"><a href="Chapter_38.html#statistical-tests"><i class="fa fa-check"></i><b>38.3</b> Statistical tests</a>
<ul>
<li class="chapter" data-level="38.3.1" data-path="Chapter_38.html"><a href="Chapter_38.html#reporting-t-tests"><i class="fa fa-check"></i><b>38.3.1</b> Reporting t-tests</a></li>
<li class="chapter" data-level="38.3.2" data-path="Chapter_38.html"><a href="Chapter_38.html#reporting-anova"><i class="fa fa-check"></i><b>38.3.2</b> Reporting ANOVA</a></li>
<li class="chapter" data-level="38.3.3" data-path="Chapter_38.html"><a href="Chapter_38.html#reporting-a-main-whitney-u-test"><i class="fa fa-check"></i><b>38.3.3</b> Reporting a Main-Whitney U test</a></li>
<li class="chapter" data-level="38.3.4" data-path="Chapter_38.html"><a href="Chapter_38.html#reporting-a-wilcoxon-signed-rank-test"><i class="fa fa-check"></i><b>38.3.4</b> Reporting a Wilcoxon signed-rank test</a></li>
<li class="chapter" data-level="38.3.5" data-path="Chapter_38.html"><a href="Chapter_38.html#reporting-chi-square-tests"><i class="fa fa-check"></i><b>38.3.5</b> Reporting Chi-square tests</a></li>
<li class="chapter" data-level="38.3.6" data-path="Chapter_38.html"><a href="Chapter_38.html#reporting-correlation-coefficients"><i class="fa fa-check"></i><b>38.3.6</b> Reporting correlation coefficients</a></li>
<li class="chapter" data-level="38.3.7" data-path="Chapter_38.html"><a href="Chapter_38.html#reporting-regressions"><i class="fa fa-check"></i><b>38.3.7</b> Reporting regressions</a></li>
</ul></li>
<li class="chapter" data-level="38.4" data-path="Chapter_38.html"><a href="Chapter_38.html#conclusions"><i class="fa fa-check"></i><b>38.4</b> Conclusions</a></li>
</ul></li>
<li class="chapter" data-level="39" data-path="Chapter_39.html"><a href="Chapter_39.html"><i class="fa fa-check"></i><b>39</b> <em>Practical</em>. Statistical techniques in R</a>
<ul>
<li class="chapter" data-level="39.1" data-path="Chapter_39.html"><a href="Chapter_39.html#working-with-a-data-set"><i class="fa fa-check"></i><b>39.1</b> Working with a data set</a></li>
<li class="chapter" data-level="39.2" data-path="Chapter_39.html"><a href="Chapter_39.html#familiar-statistical-tests-in-r"><i class="fa fa-check"></i><b>39.2</b> Familiar statistical tests in R</a>
<ul>
<li class="chapter" data-level="39.2.1" data-path="Chapter_39.html"><a href="Chapter_39.html#one-way-anova-in-r"><i class="fa fa-check"></i><b>39.2.1</b> One-way ANOVA in R</a></li>
<li class="chapter" data-level="39.2.2" data-path="Chapter_39.html"><a href="Chapter_39.html#two-way-anova-in-r"><i class="fa fa-check"></i><b>39.2.2</b> Two-way ANOVA in R</a></li>
<li class="chapter" data-level="39.2.3" data-path="Chapter_39.html"><a href="Chapter_39.html#chi-square-test-in-r"><i class="fa fa-check"></i><b>39.2.3</b> Chi-square test in R</a></li>
<li class="chapter" data-level="39.2.4" data-path="Chapter_39.html"><a href="Chapter_39.html#correlation-test-in-r"><i class="fa fa-check"></i><b>39.2.4</b> Correlation test in R</a></li>
<li class="chapter" data-level="39.2.5" data-path="Chapter_39.html"><a href="Chapter_39.html#simple-linear-regression-in-r"><i class="fa fa-check"></i><b>39.2.5</b> Simple linear regression in R</a></li>
<li class="chapter" data-level="39.2.6" data-path="Chapter_39.html"><a href="Chapter_39.html#multiple-regression-in-r"><i class="fa fa-check"></i><b>39.2.6</b> Multiple regression in R</a></li>
</ul></li>
<li class="chapter" data-level="39.3" data-path="Chapter_39.html"><a href="Chapter_39.html#statistical-test-assumptions-in-r"><i class="fa fa-check"></i><b>39.3</b> Statistical test assumptions in R</a></li>
</ul></li>
<li class="appendix"><span><b>Appendix</b></span></li>
<li class="chapter" data-level="A" data-path="appendexA.html"><a href="appendexA.html"><i class="fa fa-check"></i><b>A</b> Answers to chapter exercises</a>
<ul>
<li class="chapter" data-level="A.1" data-path="appendexA.html"><a href="appendexA.html#chapter-3"><i class="fa fa-check"></i><b>A.1</b> Chapter 3</a>
<ul>
<li class="chapter" data-level="A.1.1" data-path="appendexA.html"><a href="appendexA.html#exercise-3.1"><i class="fa fa-check"></i><b>A.1.1</b> Exercise 3.1:</a></li>
<li class="chapter" data-level="A.1.2" data-path="appendexA.html"><a href="appendexA.html#exercise-3.2"><i class="fa fa-check"></i><b>A.1.2</b> Exercise 3.2</a></li>
<li class="chapter" data-level="A.1.3" data-path="appendexA.html"><a href="appendexA.html#exercise-3.2-1"><i class="fa fa-check"></i><b>A.1.3</b> Exercise 3.2</a></li>
<li class="chapter" data-level="A.1.4" data-path="appendexA.html"><a href="appendexA.html#exercise-3.4"><i class="fa fa-check"></i><b>A.1.4</b> Exercise 3.4</a></li>
</ul></li>
<li class="chapter" data-level="A.2" data-path="appendexA.html"><a href="appendexA.html#chapter-8"><i class="fa fa-check"></i><b>A.2</b> Chapter 8</a>
<ul>
<li class="chapter" data-level="A.2.1" data-path="appendexA.html"><a href="appendexA.html#exercise-8.1"><i class="fa fa-check"></i><b>A.2.1</b> Exercise 8.1</a></li>
<li class="chapter" data-level="A.2.2" data-path="appendexA.html"><a href="appendexA.html#exercise-8.2"><i class="fa fa-check"></i><b>A.2.2</b> Exercise 8.2</a></li>
<li class="chapter" data-level="A.2.3" data-path="appendexA.html"><a href="appendexA.html#exercise-8.3"><i class="fa fa-check"></i><b>A.2.3</b> Exercise 8.3</a></li>
</ul></li>
<li class="chapter" data-level="A.3" data-path="appendexA.html"><a href="appendexA.html#chapter-14"><i class="fa fa-check"></i><b>A.3</b> Chapter 14</a>
<ul>
<li class="chapter" data-level="A.3.1" data-path="appendexA.html"><a href="appendexA.html#exercise-14.1"><i class="fa fa-check"></i><b>A.3.1</b> Exercise 14.1</a></li>
<li class="chapter" data-level="A.3.2" data-path="appendexA.html"><a href="appendexA.html#exercise-14.2"><i class="fa fa-check"></i><b>A.3.2</b> Exercise 14.2</a></li>
<li class="chapter" data-level="A.3.3" data-path="appendexA.html"><a href="appendexA.html#exercise-13.3"><i class="fa fa-check"></i><b>A.3.3</b> Exercise 13.3</a></li>
<li class="chapter" data-level="A.3.4" data-path="appendexA.html"><a href="appendexA.html#exercise-13.4"><i class="fa fa-check"></i><b>A.3.4</b> Exercise 13.4</a></li>
<li class="chapter" data-level="A.3.5" data-path="appendexA.html"><a href="appendexA.html#exercise-13.5"><i class="fa fa-check"></i><b>A.3.5</b> Exercise 13.5</a></li>
</ul></li>
<li class="chapter" data-level="A.4" data-path="appendexA.html"><a href="appendexA.html#chapter-17"><i class="fa fa-check"></i><b>A.4</b> Chapter 17</a>
<ul>
<li class="chapter" data-level="A.4.1" data-path="appendexA.html"><a href="appendexA.html#exercise-17.1"><i class="fa fa-check"></i><b>A.4.1</b> Exercise 17.1</a></li>
<li class="chapter" data-level="A.4.2" data-path="appendexA.html"><a href="appendexA.html#exercise-17.2"><i class="fa fa-check"></i><b>A.4.2</b> Exercise 17.2</a></li>
<li class="chapter" data-level="A.4.3" data-path="appendexA.html"><a href="appendexA.html#exercise-17.3"><i class="fa fa-check"></i><b>A.4.3</b> Exercise 17.3</a></li>
</ul></li>
<li class="chapter" data-level="A.5" data-path="appendexA.html"><a href="appendexA.html#chapter-20"><i class="fa fa-check"></i><b>A.5</b> Chapter 20</a>
<ul>
<li class="chapter" data-level="A.5.1" data-path="appendexA.html"><a href="appendexA.html#exercise-20.1"><i class="fa fa-check"></i><b>A.5.1</b> Exercise 20.1</a></li>
<li class="chapter" data-level="A.5.2" data-path="appendexA.html"><a href="appendexA.html#exercise-20.2"><i class="fa fa-check"></i><b>A.5.2</b> Exercise 20.2</a></li>
<li class="chapter" data-level="A.5.3" data-path="appendexA.html"><a href="appendexA.html#exercise-20.3"><i class="fa fa-check"></i><b>A.5.3</b> Exercise 20.3</a></li>
<li class="chapter" data-level="A.5.4" data-path="appendexA.html"><a href="appendexA.html#exercise-20.4"><i class="fa fa-check"></i><b>A.5.4</b> Exercise 20.4</a></li>
<li class="chapter" data-level="A.5.5" data-path="appendexA.html"><a href="appendexA.html#exercise-20.5"><i class="fa fa-check"></i><b>A.5.5</b> Exercise 20.5</a></li>
</ul></li>
<li class="chapter" data-level="A.6" data-path="appendexA.html"><a href="appendexA.html#chapter-23"><i class="fa fa-check"></i><b>A.6</b> Chapter 23</a>
<ul>
<li class="chapter" data-level="A.6.1" data-path="appendexA.html"><a href="appendexA.html#exercise-23.1"><i class="fa fa-check"></i><b>A.6.1</b> Exercise 23.1</a></li>
<li class="chapter" data-level="A.6.2" data-path="appendexA.html"><a href="appendexA.html#exercise-23.2"><i class="fa fa-check"></i><b>A.6.2</b> Exercise 23.2</a></li>
<li class="chapter" data-level="A.6.3" data-path="appendexA.html"><a href="appendexA.html#exercise-23.3"><i class="fa fa-check"></i><b>A.6.3</b> Exercise 23.3</a></li>
<li class="chapter" data-level="A.6.4" data-path="appendexA.html"><a href="appendexA.html#exercise-23.4"><i class="fa fa-check"></i><b>A.6.4</b> Exercise 23.4</a></li>
<li class="chapter" data-level="A.6.5" data-path="appendexA.html"><a href="appendexA.html#exercise-23.5"><i class="fa fa-check"></i><b>A.6.5</b> Exercise 23.5</a></li>
</ul></li>
<li class="chapter" data-level="A.7" data-path="appendexA.html"><a href="appendexA.html#chapter-28"><i class="fa fa-check"></i><b>A.7</b> Chapter 28</a>
<ul>
<li class="chapter" data-level="A.7.1" data-path="appendexA.html"><a href="appendexA.html#exercise-28.1"><i class="fa fa-check"></i><b>A.7.1</b> Exercise 28.1</a></li>
<li class="chapter" data-level="A.7.2" data-path="appendexA.html"><a href="appendexA.html#exercise-28.2"><i class="fa fa-check"></i><b>A.7.2</b> Exercise 28.2</a></li>
<li class="chapter" data-level="A.7.3" data-path="appendexA.html"><a href="appendexA.html#exercise-28.3"><i class="fa fa-check"></i><b>A.7.3</b> Exercise 28.3</a></li>
<li class="chapter" data-level="A.7.4" data-path="appendexA.html"><a href="appendexA.html#exercise-28.4"><i class="fa fa-check"></i><b>A.7.4</b> Exercise 28.4</a></li>
</ul></li>
<li class="chapter" data-level="A.8" data-path="appendexA.html"><a href="appendexA.html#chapter-31"><i class="fa fa-check"></i><b>A.8</b> Chapter 31</a>
<ul>
<li class="chapter" data-level="A.8.1" data-path="appendexA.html"><a href="appendexA.html#exercise-31.1"><i class="fa fa-check"></i><b>A.8.1</b> Exercise 31.1</a></li>
<li class="chapter" data-level="A.8.2" data-path="appendexA.html"><a href="appendexA.html#exercise-31.2"><i class="fa fa-check"></i><b>A.8.2</b> Exercise 31.2</a></li>
<li class="chapter" data-level="A.8.3" data-path="appendexA.html"><a href="appendexA.html#exercise-31.3"><i class="fa fa-check"></i><b>A.8.3</b> Exercise 31.3</a></li>
</ul></li>
<li class="chapter" data-level="A.9" data-path="appendexA.html"><a href="appendexA.html#exercise-31.4"><i class="fa fa-check"></i><b>A.9</b> Exercise 31.4</a>
<ul>
<li class="chapter" data-level="A.9.1" data-path="appendexA.html"><a href="appendexA.html#exercise-31.5"><i class="fa fa-check"></i><b>A.9.1</b> Exercise 31.5</a></li>
</ul></li>
<li class="chapter" data-level="A.10" data-path="appendexA.html"><a href="appendexA.html#chapter-34"><i class="fa fa-check"></i><b>A.10</b> Chapter 34</a>
<ul>
<li class="chapter" data-level="A.10.1" data-path="appendexA.html"><a href="appendexA.html#exercise-34.1"><i class="fa fa-check"></i><b>A.10.1</b> Exercise 34.1</a></li>
<li class="chapter" data-level="A.10.2" data-path="appendexA.html"><a href="appendexA.html#exercise-34.2"><i class="fa fa-check"></i><b>A.10.2</b> Exercise 34.2</a></li>
<li class="chapter" data-level="A.10.3" data-path="appendexA.html"><a href="appendexA.html#exercise-34.3"><i class="fa fa-check"></i><b>A.10.3</b> Exercise 34.3</a></li>
<li class="chapter" data-level="A.10.4" data-path="appendexA.html"><a href="appendexA.html#exercise-34.4"><i class="fa fa-check"></i><b>A.10.4</b> Exercise 34.4</a></li>
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<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">Fundamental statistical concepts and techniques in the biological and environmental sciences: With jamovi</a>
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<div id="Chapter_36" class="section level1 hasAnchor" number="36">
<h1><span class="header-section-number">Chapter 36</span> <em>Practical</em>. Introduction to R<a href="Chapter_36.html#Chapter_36" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>Throughout this book, we have used jamovi for statistical analysis.
Jamovi is excellent software for running simple statistical analyses, but it has its limitations.
Jamovi is built upon the programming language R, which is much more powerful and versatile in comparison.
The R programming language is now widely used among scientists for data analysis.
It can be used to analyse and plot data, run computer simulations, or even write slides, papers, or books (this book was written using R).
The R programming language is completely free and open source, as is the popular <a href="https://posit.co/downloads/">Rstudio</a> software for using it.
The R programming language specialises in statistical computing, which is part of the reason for its popularity among scientists.</p>
<p>Another reason for the popularity of R is its versatility, and the ease with which new techniques can be shared.
Imagine that you develop a new method for analysing data.
If you want other researchers to be able to use your method in their research, then you could write your own software from scratch for them to install and use.
But doing this would be very time consuming, and a lot of that time would likely be spent writing the graphical user interface and making sure that your program worked across platforms (e.g., on Windows and Mac).
Worse, once written, there would be no easy way to make your program work with other statistical software should you need to integrate different analyses or visualisation tools (e.g., plotting data).
To avoid all of this, you could instead just present your new method for data analysis and let other researchers write their own code for implementing it.
But not all researchers will have the time or expertise to write their own code.</p>
<p>Instead, R allows researchers to write new tools for data analysis using simple coding scripts.
These scripts are organised into R packages, which can be uploaded by authors to the Comprehensive R Archive Network (CRAN)<a href="#fn88" class="footnote-ref" id="fnref88"><sup>88</sup></a>, then downloaded by users with a single command in R.
This way, there is no need for completely different software to be used for different analyses; all analyses can be written and run in R.</p>
<p>The downside to all of this is that learning R can be a bit daunting at first.
Running analyses is not done by pointing and clicking on icons as in jamovi.
You need to use code.
This practical will start with the very basics and work up to some simple data analyses.
Rather than questions to answer, this practical has tasks for you to try to complete in R.</p>
<div id="getting-used-to-the-r-interface" class="section level2 hasAnchor" number="36.1">
<h2><span class="header-section-number">36.1</span> Getting used to the R interface<a href="Chapter_36.html#getting-used-to-the-r-interface" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>There are two ways that you can use R.
The first way is to download R and Rstudio on your own computer.
Both R and Rstudio are free to download and use, and work on any major operating system (Windows, Mac, and Linux).
To download R, go to the Comprehensive R Archive Network (<a href="https://cran.r-project.org/" class="uri">https://cran.r-project.org/</a>) and follow the instructions for your operating system (OS).
To download Rstudio, go to the Posit website<a href="#fn89" class="footnote-ref" id="fnref89"><sup>89</sup></a> and choose the appropriate download for your operating system.</p>
<p>If you cannot or do not want to download R and Rstudio on your own computer, then you can still use both by going to the Rstudio Cloud (<a href="https://rstudio.cloud/" class="uri">https://rstudio.cloud/</a>) and running Rstudio from a browser such as Firefox or Chrome.
You will need to click the green “Get Started” button and sign up for a free account.
When you open Rstudio either on your own computer or the cloud, you will see several windows open, including a console.
The console will look something like the below.</p>
<pre><code>R version 4.2.2 Patched (2022-11-10 r83330) -- "Innocent and Trusting"
Copyright (C) 2022 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)
R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.
Natural language support but running in an English locale
R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.
Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.
> </code></pre>
<p>If you click to the right of the greater than sign <code>></code>, you can start using R right in the console.
You can use R as a standard calculator here to get a feel for the console.
Try typing something like the below (semi-colons are not actually necessary).
For example, you could add 2 + 5.</p>
<div class="sourceCode" id="cb121"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb121-1"><a href="Chapter_36.html#cb121-1" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span> <span class="sc">+</span> <span class="dv">5</span>; </span></code></pre></div>
<pre><code>[1] 7</code></pre>
<p>Try this yourself.</p>
<blockquote>
<p><strong>Task 1: Add 2 numbers together in R</strong></p>
</blockquote>
<p>You can also multiply in R.</p>
<div class="sourceCode" id="cb123"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb123-1"><a href="Chapter_36.html#cb123-1" aria-hidden="true" tabindex="-1"></a><span class="dv">4</span> <span class="sc">*</span> <span class="dv">4</span>; </span></code></pre></div>
<pre><code>[1] 16</code></pre>
<p>The caret key <code>^</code> is used for exponents</p>
<div class="sourceCode" id="cb125"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb125-1"><a href="Chapter_36.html#cb125-1" aria-hidden="true" tabindex="-1"></a><span class="dv">5</span><span class="sc">^</span><span class="dv">2</span>; </span></code></pre></div>
<pre><code>[1] 25</code></pre>
<p>R will use the correct order of operations (see <a href="Chapter_1.html#order-of-operations">Chapter 1.3</a>) when doing mathematical calculations.
For example, it will know to calculate the exponent before multiplying, and to multiply before adding.</p>
<div class="sourceCode" id="cb127"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb127-1"><a href="Chapter_36.html#cb127-1" aria-hidden="true" tabindex="-1"></a><span class="dv">2</span> <span class="sc">+</span> <span class="dv">4</span> <span class="sc">*</span> <span class="dv">5</span><span class="sc">^</span><span class="dv">2</span>; </span></code></pre></div>
<pre><code>[1] 102</code></pre>
<p>Following the correct order of operations, <span class="math inline">\(5^{2} = 25\)</span>, which is then multiplied by 4 to get <span class="math inline">\(4 \times 25 = 100\)</span>, and we add 2 to get <span class="math inline">\(100 + 2 = 102\)</span>.
We can, however, use parentheses to specify a different order of operations.</p>
<div class="sourceCode" id="cb129"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb129-1"><a href="Chapter_36.html#cb129-1" aria-hidden="true" tabindex="-1"></a>(<span class="dv">2</span> <span class="sc">+</span> <span class="dv">4</span>) <span class="sc">*</span> <span class="dv">5</span><span class="sc">^</span><span class="dv">2</span>; </span></code></pre></div>
<pre><code>[1] 150</code></pre>
<p>Now R will calculate <span class="math inline">\(2 + 4 = 6\)</span> and multiply the 6 by 25 to get an answer of 150.
The R console functions very well as a calculator.
Instead of punching buttons into a hand calculator or mobile phone, an entire equation can be written and placed into the console.
This makes mistakes less likely.
For example, in <a href="Chapter_1.html#order-of-operations">Chapter 1.3</a>, the following equation was presented,</p>
<p><span class="math display">\[x = 3^{2} + 2(1 + 3)^{2} - 6 \times 0.\]</span></p>
<p>To solve this in R, we just need to type the full equation in the console.</p>
<div class="sourceCode" id="cb131"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb131-1"><a href="Chapter_36.html#cb131-1" aria-hidden="true" tabindex="-1"></a><span class="dv">3</span><span class="sc">^</span><span class="dv">2</span> <span class="sc">+</span> <span class="dv">2</span><span class="sc">*</span>(<span class="dv">1</span> <span class="sc">+</span> <span class="dv">3</span>)<span class="sc">^</span><span class="dv">2</span> <span class="sc">-</span> <span class="dv">6</span> <span class="sc">*</span> <span class="dv">0</span>;</span></code></pre></div>
<pre><code>[1] 41</code></pre>
<p>We get the correct answer of 41.
Note that there needed to be an asterisk (<code>*</code>) to indicate multiplication between the 2 and the left parentheses.
This is because parentheses specify specific functions in R.
We will introduce functions next, but first, try the following task.</p>
<blockquote>
<p><strong>Task 2: Use the console to calculate <span class="math inline">\(x = \frac{2^2 + 1}{3^2 + 2}\)</span> (hint, you need to put the top and bottom of the fraction in parentheses, e.g., <code>(2^2 + 1)</code> for the numerator, and use the forward slash <code>/</code> for division).</strong></p>
</blockquote>
<p>You should get an answer of 0.4545455.</p>
<p>As previously mentioned, parentheses specify functions in R.
Functions have specific names such as <code>sqrt</code>, which calculates the square root of anything within the parentheses <code>sqrt()</code>.
If, for example, you wanted to find the square root of some number, you could use the <code>sqrt</code> function below.</p>
<div class="sourceCode" id="cb133"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb133-1"><a href="Chapter_36.html#cb133-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sqrt</span>(<span class="dv">256</span>);</span></code></pre></div>
<pre><code>[1] 16</code></pre>
<p>The console returns the correct answer 16.
Similar functions exist for logarithms (<code>log</code>) and trigonometric functions (e.g., <code>sin</code>, <code>cos</code>).
The parentheses after the word indicate that some function is being called in R.
Try to use the square root function to solve for <span class="math inline">\(x\)</span> in another equation.</p>
<blockquote>
<p><strong>Task 3: Use the console to calculate <span class="math inline">\(\sqrt{3 + 4^2}\)</span>.</strong></p>
</blockquote>
<p>You should get an answer of 4.3588989.</p>
<p>If you type something into the console incorrectly, you do not need to retype it entirely.
You can use the up arrow to scroll through the history of console input.
Try doing that now.</p>
<blockquote>
<p><strong>Task 4: Use the up arrow to find your previous calculation for <span class="math inline">\(\sqrt{3 + 4^2}\)</span>, then change this slightly to calculate <span class="math inline">\(\sqrt{3 + 4^3}\)</span>.</strong></p>
</blockquote>
<p>You should get an answer of 8.1853528.</p>
<p>Functions do not need to be mathematical like the <code>sqrt</code> function.
For example, the <code>getwd</code> function can be used to let you know what directory (i.e., folder) you are working in.</p>
<div class="sourceCode" id="cb135"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb135-1"><a href="Chapter_36.html#cb135-1" aria-hidden="true" tabindex="-1"></a><span class="fu">getwd</span>();</span></code></pre></div>
<pre><code>[1] "/home/brad/Dropbox/teaching/statistics_book/statistics_book"</code></pre>
<p>If we were to save or load a file within R, this is the location on the computer from which R would try to load.
We could also use the <code>setwd</code> function to set a new working directory (type the working directory in quotes inside the parentheses: <code>setwd("folder/subfolder/etc")</code>).
You can also set the working directory in Rstudio by going to the toolbar and clicking ‘Session’ and selecting ‘Set Working Directory’.</p>
</div>
<div id="assigning-variables-in-the-r-console" class="section level2 hasAnchor" number="36.2">
<h2><span class="header-section-number">36.2</span> Assigning variables in the R console<a href="Chapter_36.html#assigning-variables-in-the-r-console" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>In R, we can also assign values to variables using the characters <code><-</code> to make an arrow.
For example, we might want to set <code>var_1</code> equal 10.</p>
<div class="sourceCode" id="cb137"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb137-1"><a href="Chapter_36.html#cb137-1" aria-hidden="true" tabindex="-1"></a>var_1 <span class="ot"><-</span> <span class="dv">10</span>;</span></code></pre></div>
<p>We can now use <code>var_1</code> in the console.
For example, since <code>var_1</code> now equals 10, if we multiply <code>var_1</code> by 5, R calculates <span class="math inline">\(10 \times 5 = 50\)</span>.</p>
<div class="sourceCode" id="cb138"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb138-1"><a href="Chapter_36.html#cb138-1" aria-hidden="true" tabindex="-1"></a>var_1 <span class="sc">*</span> <span class="dv">5</span>; <span class="co"># Multiplying the variable by 5</span></span></code></pre></div>
<pre><code>[1] 50</code></pre>
<p>The correct value of 50 is returned because <code>var_1</code> equals 10. Also note the comment left after the <code>#</code> key.
In R, anything that comes after <code>#</code> on a line is a comment that R ignores.
Comments are ways of explaining in plain words what the code is doing, or drawing attention to important notes about the code.</p>
<blockquote>
<p><strong>Task 5: Assign a new variable to a value of 4 (call it anything you want), then multiply the new variable by itself.</strong></p>
</blockquote>
<p>For Task 5, you should get an output of 16.</p>
<p>When we assign something to a variable like <code>var_1</code>, we are not limited to a single number.
Variables in R can be much more complex.
For example, we can assign a new variable <code>vector_1</code> to an ordered set of 6 different numbers using the <code>c</code> function.
The <code>c</code> function combines values together.</p>
<div class="sourceCode" id="cb140"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb140-1"><a href="Chapter_36.html#cb140-1" aria-hidden="true" tabindex="-1"></a>vector_1 <span class="ot"><-</span> <span class="fu">c</span>(<span class="dv">5</span>, <span class="dv">1</span>, <span class="dv">3</span>, <span class="dv">5</span>, <span class="dv">7</span>, <span class="dv">11</span>); <span class="co"># Six numbers</span></span>
<span id="cb140-2"><a href="Chapter_36.html#cb140-2" aria-hidden="true" tabindex="-1"></a>vector_1; <span class="co"># Prints out the vector</span></span></code></pre></div>
<pre><code>[1] 5 1 3 5 7 11</code></pre>
<p>Doing this makes it possible to calculate for all of the values in <code>vector_1</code> at the same time.
We might, for example, want to multiply all of the numbers by 10.</p>
<div class="sourceCode" id="cb142"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb142-1"><a href="Chapter_36.html#cb142-1" aria-hidden="true" tabindex="-1"></a>vector_1 <span class="sc">*</span> <span class="dv">10</span></span></code></pre></div>
<pre><code>[1] 50 10 30 50 70 110</code></pre>
<p>Variables also do not necessarily need to be numbers.
We might assign a new variable <code>phrase_1</code> to a string of letters.
When doing this, the letters need to be enclosed in quotation marks.</p>
<div class="sourceCode" id="cb144"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb144-1"><a href="Chapter_36.html#cb144-1" aria-hidden="true" tabindex="-1"></a>phrase_1 <span class="ot"><-</span> <span class="st">"string of words"</span>;</span>
<span id="cb144-2"><a href="Chapter_36.html#cb144-2" aria-hidden="true" tabindex="-1"></a>phrase_1;</span></code></pre></div>
<pre><code>[1] "string of words"</code></pre>
<p>We can even combine numbers, vectors, and words into a single object as a list using the <code>list</code> function.</p>
<div class="sourceCode" id="cb146"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb146-1"><a href="Chapter_36.html#cb146-1" aria-hidden="true" tabindex="-1"></a>object_1 <span class="ot"><-</span> <span class="fu">list</span>(var_1, vector_1, phrase_1);</span>
<span id="cb146-2"><a href="Chapter_36.html#cb146-2" aria-hidden="true" tabindex="-1"></a>object_1;</span></code></pre></div>
<pre><code>[[1]]
[1] 10
[[2]]
[1] 5 1 3 5 7 11
[[3]]
[1] "string of words"</code></pre>
<p>There are far too many possibilities to introduce everything about how R works.
The best way to get started is to play around and see what works and what does not.
You will not break anything.
Error messages are good because they can help you learn how R works.</p>
<blockquote>
<p><strong>Task 6: Use the R console until you find a new error message.</strong></p>
</blockquote>
<p>Try something new in the R console.
Keep going until you get an error message that you have not yet seen, then move on to the next exercise.</p>
</div>
<div id="some-descriptive-statistics" class="section level2 hasAnchor" number="36.3">
<h2><span class="header-section-number">36.3</span> Some descriptive statistics<a href="Chapter_36.html#some-descriptive-statistics" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Now we can get started with some statistical analyses.
In <a href="Chapter_39.html#Chapter_39">Chapter 39</a> we will learn how to do some familiar analyses with R scripts and data uploaded from CSV files, but for now, we will just type our data directly into the console.
We can use the data from <a href="Chapter_35.html#randomisation-for-hypothesis-testing">Chapter 35.4</a> as an example.
The code below reads in 2 different variables, <code>SO1</code> and <code>SO2</code>.
The numbers are the same ovipositor lengths from the histograms in Figure 34.2 of <a href="Chapter_35.html#randomisation-for-hypothesis-testing">Chapter 35.4</a>.</p>
<div class="sourceCode" id="cb148"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb148-1"><a href="Chapter_36.html#cb148-1" aria-hidden="true" tabindex="-1"></a>SO1 <span class="ot"><-</span> <span class="fu">c</span>(<span class="fl">3.256</span>, <span class="fl">3.133</span>, <span class="fl">3.071</span>, <span class="fl">2.299</span>, <span class="fl">2.995</span>, <span class="fl">2.929</span>, <span class="fl">3.291</span>, <span class="fl">2.658</span>, <span class="fl">3.406</span>, </span>
<span id="cb148-2"><a href="Chapter_36.html#cb148-2" aria-hidden="true" tabindex="-1"></a> <span class="fl">2.976</span>, <span class="fl">2.817</span>, <span class="fl">3.133</span>, <span class="fl">3.000</span>, <span class="fl">3.027</span>, <span class="fl">3.178</span>, <span class="fl">3.133</span>, <span class="fl">3.210</span>);</span>
<span id="cb148-3"><a href="Chapter_36.html#cb148-3" aria-hidden="true" tabindex="-1"></a>SO2 <span class="ot"><-</span> <span class="fu">c</span>(<span class="fl">3.014</span>, <span class="fl">2.790</span>, <span class="fl">2.985</span>, <span class="fl">2.911</span>, <span class="fl">2.914</span>, <span class="fl">2.724</span>, <span class="fl">2.967</span>, <span class="fl">2.745</span>, <span class="fl">2.973</span>, </span>
<span id="cb148-4"><a href="Chapter_36.html#cb148-4" aria-hidden="true" tabindex="-1"></a> <span class="fl">2.560</span>, <span class="fl">2.837</span>, <span class="fl">2.883</span>, <span class="fl">2.668</span>, <span class="fl">3.063</span>, <span class="fl">2.639</span>);</span></code></pre></div>
<p>Copy and paste the code above into the R console.
You should now have 2 new variables, <code>SO1</code> and <code>SO2</code></p>
<blockquote>
<p><strong>Task 7: Type <code>SO1</code> into the R console to print the ovipositor lengths for species SO1, then do the same for <code>SO2</code></strong>.</p>
</blockquote>
<p>We can replicate the histogram from Figure 35.2A in <a href="Chapter_35.html#randomisation-for-hypothesis-testing">Chapter 35.4</a> with the following code.</p>
<div class="sourceCode" id="cb149"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb149-1"><a href="Chapter_36.html#cb149-1" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(<span class="at">x =</span> SO1, <span class="at">xlab =</span> <span class="st">"SO1 ovipositor length (mm)"</span>, <span class="at">main =</span> <span class="st">""</span>);</span></code></pre></div>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-210"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-210-1.png" alt="A histogram of SO1 ovipositor lengths that is roughly normally distributed" width="672" />
<p class="caption">
Figure 36.1: Ovipositor length distributions for an unnamed species of fig wasps SO1.
</p>
</div>
<p>The bin widths in Figure 36.1 are not the same as Figure 35.2A because there are other options within <code>hist</code> that have not been specified.
Within <code>hist</code> and other functions, these options such as <code>x</code>, <code>xlab</code>, and <code>main</code> are called <em>arguments</em>.
The word ‘argument’ in this case has nothing to do with a disagreement or logical reasoning.
In this case, it just refers to information that is passed in a function.</p>
<blockquote>
<p><strong>Task 8: Make a histogram of SO2 ovipositor lengths. Try using the argument <code>col = "blue"</code> in the <code>hist</code> function to make the histogram bars blue.</strong></p>
</blockquote>
<p>Now we can try collecting some summary statistics for SO1 and SO2.
In the lab practical from <a href="Chapter_14.html#Chapter_14">Chapter 14</a>, we used jamovi to calculate summary statistics.
Table 36.1 below shows how those same summary statistics can be calculated using functions in R, and the output of each function for SO1.</p>
<table>
<caption><span id="tab:unnamed-chunk-211">Table 36.1: </span>R functions and output for summary statistics applied to a variable (SO1) of ovipositor lengths in an unnamed species of nonpollinating fig wasp.</caption>
<thead>
<tr class="header">
<th align="left">Statistic</th>
<th align="left">Function</th>
<th align="left">Output</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">N</td>
<td align="left"><code>length(SO1)</code></td>
<td align="left">17</td>
</tr>
<tr class="even">
<td align="left">Std. deviation</td>
<td align="left"><code>sd(SO1)</code></td>
<td align="left">0.260018</td>
</tr>
<tr class="odd">
<td align="left">Variance</td>
<td align="left"><code>var(SO1)</code></td>
<td align="left">0.067609</td>
</tr>
<tr class="even">
<td align="left">Minimum</td>
<td align="left"><code>min(SO1)</code></td>
<td align="left">2.299</td>
</tr>
<tr class="odd">
<td align="left">Maximum</td>
<td align="left"><code>max(SO1)</code></td>
<td align="left">3.406</td>
</tr>
<tr class="even">
<td align="left">Range</td>
<td align="left"><code>range(SO1)</code></td>
<td align="left">2.299, 3.406</td>
</tr>
<tr class="odd">
<td align="left">IQR</td>
<td align="left"><code>IQR(SO1)</code></td>
<td align="left">0.202</td>
</tr>
<tr class="even">
<td align="left">Mean</td>
<td align="left"><code>mean(SO1)</code></td>
<td align="left">3.030118</td>
</tr>
<tr class="odd">
<td align="left">Median</td>
<td align="left"><code>median(SO1)</code></td>
<td align="left">3.071</td>
</tr>
</tbody>
</table>
<p>Notice that the <code>range</code> function gives the minimum and maximum of SO1, so we need to subtract the latter from the former to get the actual range.</p>
<blockquote>
<p><strong>Task 9: Calculate the summary statistics in Table 35.1 for SO2.</strong></p>
</blockquote>
<p>There is no function for standard error in R.
We could write a custom function to calculate the standard error, but for now we can just use the formula from <a href="Chapter_12.html#the-standard-error">Chapter 12.6</a> to calculate the standard error of SO1,</p>
<p><span class="math display">\[SE = \frac{s}{\sqrt{N}}.\]</span></p>
<p>Since we have the standard deviation (<span class="math inline">\(s\)</span>) and sample size (<span class="math inline">\(N\)</span>) from Table 36.1, we can calculate SE in R,</p>
<div class="sourceCode" id="cb150"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb150-1"><a href="Chapter_36.html#cb150-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sd</span>(SO1) <span class="sc">/</span> <span class="fu">sqrt</span>( <span class="fu">length</span>(SO1) );</span></code></pre></div>
<pre><code>[1] 0.06306363</code></pre>
<p>The code above calculates the standard deviation of SO1 (<code>sd(SO1)</code>), then divides (<code>/</code>) by the square root (<code>sqrt()</code>) of the length of SO1 (<code>length(SO1)</code>).
This can be difficult to read because <code>length(SO1)</code> is enclosed in <code>sqrt()</code>.
But we can break this down step by step to make it easier to read by assigning each component to a new variable.</p>
<div class="sourceCode" id="cb152"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb152-1"><a href="Chapter_36.html#cb152-1" aria-hidden="true" tabindex="-1"></a>SO1_SD <span class="ot"><-</span> <span class="fu">sd</span>(SO1); <span class="co"># Assign the standard deviation</span></span>
<span id="cb152-2"><a href="Chapter_36.html#cb152-2" aria-hidden="true" tabindex="-1"></a>SO1_N <span class="ot"><-</span> <span class="fu">length</span>(SO1); <span class="co"># Assign the N</span></span>
<span id="cb152-3"><a href="Chapter_36.html#cb152-3" aria-hidden="true" tabindex="-1"></a>SO1_SD <span class="sc">/</span> <span class="fu">sqrt</span>(SO1_N); <span class="co"># Do the calculation for SE</span></span></code></pre></div>
<pre><code>[1] 0.06306363</code></pre>
<p>We can do the same for SO2.</p>
<blockquote>
<p><strong>Task 10: Calculate the standard error of SO2 ovipositor lengths.</strong></p>
</blockquote>
<p>With the standard error now calculated, we can also calculate 95 per cent confidence intervals using the formula from <a href="Chapter_18.html#Chapter_18">Chapter 18</a>, just as we did for SO1 in <a href="Chapter_35.html#bootstrapping">Chapter 35.5</a>,</p>
<p><span class="math display">\[LCI = 3.03 - \left(2.120 \times \frac{0.26}{\sqrt{17}}\right),\]</span></p>
<p><span class="math display">\[UCI = 3.03 + \left(2.120 \times \frac{0.26}{\sqrt{17}}\right).\]</span></p>
<p>Note that 3.03 is the mean from Table 36.1, and <span class="math inline">\(0.26/\sqrt{17}\)</span> is the standard error that we just calculated for SO1.
The value 2.120 is the t-score associated with df = 17 - 1, i.e., 16 degrees of freedom (see <a href="Chapter_35.html#bootstrapping">Chapter 35.5</a>).
As in <a href="Chapter_35.html#bootstrapping">Chapter 35.5</a>, we get values of LCI = 2.896 and UCI = 3.164.</p>
<blockquote>
<p><strong>Task 10: Calculate 95 per cent confidence intervals for SO2 ovipositor lengths (hint: the appropriate t value for df = 14 is 2.145 instead of 2.120).</strong></p>
</blockquote>
<p>In the last exercise, we will attempt to bootstrap these confidence intervals using the method introduced in <a href="Chapter_35.html#bootstrapping">Chapter 35.5</a>.</p>
</div>
<div id="bootstrapping-confidence-intervals" class="section level2 hasAnchor" number="36.4">
<h2><span class="header-section-number">36.4</span> Bootstrapping confidence intervals<a href="Chapter_36.html#bootstrapping-confidence-intervals" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Writing resampling and randomisation procedures in R requires some knowledge of coding.
For now, all that you need to do is copy a pre-defined function into R and use it like any other function.
The function is called <code>simpleboot</code>.</p>
<div class="sourceCode" id="cb154"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb154-1"><a href="Chapter_36.html#cb154-1" aria-hidden="true" tabindex="-1"></a>simpleboot <span class="ot"><-</span> <span class="cf">function</span>(x, <span class="at">replicates =</span> <span class="dv">1000</span>, <span class="at">CIs =</span> <span class="fl">0.95</span>){</span>
<span id="cb154-2"><a href="Chapter_36.html#cb154-2" aria-hidden="true" tabindex="-1"></a> alpha <span class="ot"><-</span> <span class="dv">1</span> <span class="sc">-</span> CIs;</span>
<span id="cb154-3"><a href="Chapter_36.html#cb154-3" aria-hidden="true" tabindex="-1"></a> vals <span class="ot"><-</span> <span class="cn">NULL</span>; </span>
<span id="cb154-4"><a href="Chapter_36.html#cb154-4" aria-hidden="true" tabindex="-1"></a> i <span class="ot"><-</span> <span class="dv">0</span>; </span>
<span id="cb154-5"><a href="Chapter_36.html#cb154-5" aria-hidden="true" tabindex="-1"></a> <span class="cf">while</span>(i <span class="sc"><</span> replicates){ </span>
<span id="cb154-6"><a href="Chapter_36.html#cb154-6" aria-hidden="true" tabindex="-1"></a> boot <span class="ot"><-</span> <span class="fu">sample</span>(<span class="at">x =</span> x, <span class="at">size =</span> <span class="fu">length</span>(x), <span class="at">replace =</span> <span class="cn">TRUE</span>); </span>
<span id="cb154-7"><a href="Chapter_36.html#cb154-7" aria-hidden="true" tabindex="-1"></a> strap <span class="ot"><-</span> <span class="fu">mean</span>(boot); </span>
<span id="cb154-8"><a href="Chapter_36.html#cb154-8" aria-hidden="true" tabindex="-1"></a> vals <span class="ot"><-</span> <span class="fu">c</span>(vals, strap); </span>
<span id="cb154-9"><a href="Chapter_36.html#cb154-9" aria-hidden="true" tabindex="-1"></a> i <span class="ot"><-</span> i <span class="sc">+</span> <span class="dv">1</span>; </span>
<span id="cb154-10"><a href="Chapter_36.html#cb154-10" aria-hidden="true" tabindex="-1"></a> } </span>
<span id="cb154-11"><a href="Chapter_36.html#cb154-11" aria-hidden="true" tabindex="-1"></a> vals <span class="ot"><-</span> <span class="fu">sort</span>(<span class="at">x =</span> vals, <span class="at">decreasing =</span> <span class="cn">FALSE</span>); </span>
<span id="cb154-12"><a href="Chapter_36.html#cb154-12" aria-hidden="true" tabindex="-1"></a> lowCI <span class="ot"><-</span> vals[<span class="fu">round</span>((alpha<span class="sc">*</span><span class="fl">0.5</span>)<span class="sc">*</span>replicates)]; </span>
<span id="cb154-13"><a href="Chapter_36.html#cb154-13" aria-hidden="true" tabindex="-1"></a> highCI <span class="ot"><-</span> vals[<span class="fu">round</span>((<span class="dv">1</span><span class="sc">-</span>(alpha<span class="sc">*</span><span class="fl">0.5</span>))<span class="sc">*</span>replicates)]; </span>
<span id="cb154-14"><a href="Chapter_36.html#cb154-14" aria-hidden="true" tabindex="-1"></a> confid <span class="ot"><-</span> <span class="fu">c</span>(lowCI, highCI); </span>
<span id="cb154-15"><a href="Chapter_36.html#cb154-15" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span>(<span class="fu">list</span>(vals, confid)); </span>
<span id="cb154-16"><a href="Chapter_36.html#cb154-16" aria-hidden="true" tabindex="-1"></a>} </span></code></pre></div>
<p>The <code>simpleboot</code> function takes a variable (<code>x</code>) as an argument, bootstraps mean values <code>replicates</code> times, and produces confidence intervals at a level <code>CIs</code>.
To use it, highlight and copy the entire function, then put it into the R console.
Make sure that every line from <code>simpleboot</code> to the last <code>}</code> is copied.
Once it has been placed into the R console, it can be used just like any other function.
For example, we can run <code>simpleboot</code> and store the output for SO1 values.</p>
<div class="sourceCode" id="cb155"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb155-1"><a href="Chapter_36.html#cb155-1" aria-hidden="true" tabindex="-1"></a>SO1_95s <span class="ot"><-</span> <span class="fu">simpleboot</span>(<span class="at">x =</span> SO1, <span class="at">replicates =</span> <span class="dv">1000</span>, <span class="at">CIs =</span> <span class="fl">0.95</span>);</span></code></pre></div>
<p>The output will include a list of 2 different objects.
The first object <code>SO1_95s[[1]]</code> will be a list of 1000 bootstrapped mean values, sorted from lowest to highest.
The second object <code>SO1_95s[[2]]</code> will be the bootstrapped upper and lower confidence intervals.
First, we can take a look at a histogram of the bootstrapped mean values.</p>
<div class="sourceCode" id="cb156"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb156-1"><a href="Chapter_36.html#cb156-1" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(<span class="at">x =</span> SO1_95s[[<span class="dv">1</span>]], <span class="at">xlab =</span> <span class="st">"Bootstrapped SO1 means"</span>, <span class="at">main =</span> <span class="st">""</span>);</span></code></pre></div>
<div class="figure"><span style="display:block;" id="fig:unnamed-chunk-216"></span>
<img src="bookdown-demo_files/figure-html/unnamed-chunk-216-1.png" alt="A histogram bootstrapped mean SO1 values that appears to be roughly normally distributed" width="672" />
<p class="caption">
Figure 36.2: Bootstrapped mean values for 17 ovipositor lengths in an unnamed species of fig wasp collected in Baja, Mexico.
</p>
</div>
<p>We can then print out the lower and upper confidence intervals, which report the value of the 2.5 per cent rank and 97.5 rank bootstrapped means, respectively.</p>
<div class="sourceCode" id="cb157"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb157-1"><a href="Chapter_36.html#cb157-1" aria-hidden="true" tabindex="-1"></a><span class="fu">print</span>(SO1_95s[[<span class="dv">2</span>]]);</span></code></pre></div>
<pre><code>[1] 2.894941 3.136000</code></pre>
<p>Note that these values are not too far off the values of LCI = 2.896 and UCI = 3.164 calculated in the usual way in the previous exercise.</p>
<blockquote>
<p><strong>Task 11: Use the <code>simpleboot</code> function to calculate 95 per cent confidence intervals for SO2 mean ovipositor lengths.</strong></p>
</blockquote>
<p>The practical in <a href="Chapter_39.html#Chapter_39">Chapter 39</a> will demonstrate how to read CSV files into R and run hypothesis tests, including t-tests, ANOVAs, chi-square tests, correlation tests, and linear regression.</p>
</div>
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