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<!DOCTYPE html>
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<title>Chapter 33 Adjusted coefficient of determination | Statistical Techniques for Biological and Environmental Sciences</title>
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<meta name="twitter:title" content="Chapter 33 Adjusted coefficient of determination | Statistical Techniques for Biological and Environmental Sciences" />
<meta name="twitter:description" content="This is a lab book for the University of Stirling second year undergraduate Biological and Environmental Sciences statistics module." />
<meta name="author" content="Brad Duthie" />
<meta name="date" content="2023-03-26" />
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Preface</a>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html#why-this-module-is-important"><i class="fa fa-check"></i>Why this module is important</a></li>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html#jamovi"><i class="fa fa-check"></i>Jamovi statistical software</a></li>
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<li class="part"><span><b>I Background mathematics and data organisation</b></span></li>
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<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>
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<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>
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<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>
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<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> Practical: 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>Week 2 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>
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<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>
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<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>
<li class="chapter" data-level="7.3" data-path="Chapter_7.html"><a href="Chapter_7.html#applying-formulas-for-combining-errors"><i class="fa fa-check"></i><b>7.3</b> Applying formulas for combining errors</a></li>
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<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> Practical. 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>
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<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>Week 3 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>
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<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>
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<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>
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<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="Chapter_13.html"><a href="Chapter_13.html"><i class="fa fa-check"></i><b>13</b> <em>Practical</em>. Plotting and statistical summaries in Jamovi</a>
<ul>
<li class="chapter" data-level="13.1" data-path="Chapter_13.html"><a href="Chapter_13.html#reorganise-the-dataset-into-a-tidy-format"><i class="fa fa-check"></i><b>13.1</b> Reorganise the dataset into a tidy format</a></li>
<li class="chapter" data-level="13.2" data-path="Chapter_13.html"><a href="Chapter_13.html#histograms-and-box-whisker-plots"><i class="fa fa-check"></i><b>13.2</b> Histograms and box-whisker plots</a></li>
<li class="chapter" data-level="13.3" data-path="Chapter_13.html"><a href="Chapter_13.html#calculate-summary-statistics"><i class="fa fa-check"></i><b>13.3</b> Calculate summary statistics</a></li>
<li class="chapter" data-level="13.4" data-path="Chapter_13.html"><a href="Chapter_13.html#reporting-decimals-and-significant-figures"><i class="fa fa-check"></i><b>13.4</b> Reporting decimals and significant figures</a></li>
<li class="chapter" data-level="13.5" data-path="Chapter_13.html"><a href="Chapter_13.html#comparing-across-sites"><i class="fa fa-check"></i><b>13.5</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>Week 4 Overview</a></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> Introduction to probability models</a>
<ul>
<li class="chapter" data-level="14.1" data-path="Chapter_14.html"><a href="Chapter_14.html#an-instructive-example"><i class="fa fa-check"></i><b>14.1</b> An instructive example</a></li>
<li class="chapter" data-level="14.2" data-path="Chapter_14.html"><a href="Chapter_14.html#biological-applications"><i class="fa fa-check"></i><b>14.2</b> Biological applications</a></li>
<li class="chapter" data-level="14.3" data-path="Chapter_14.html"><a href="Chapter_14.html#sampling-with-and-without-replacement"><i class="fa fa-check"></i><b>14.3</b> Sampling with and without replacement</a></li>
<li class="chapter" data-level="14.4" data-path="Chapter_14.html"><a href="Chapter_14.html#probability-distributions"><i class="fa fa-check"></i><b>14.4</b> Probability distributions</a>
<ul>
<li class="chapter" data-level="14.4.1" data-path="Chapter_14.html"><a href="Chapter_14.html#binomial-distribution"><i class="fa fa-check"></i><b>14.4.1</b> Binomial distribution</a></li>
<li class="chapter" data-level="14.4.2" data-path="Chapter_14.html"><a href="Chapter_14.html#poisson-distribution"><i class="fa fa-check"></i><b>14.4.2</b> Poisson distribution</a></li>
<li class="chapter" data-level="14.4.3" data-path="Chapter_14.html"><a href="Chapter_14.html#uniform-distribution"><i class="fa fa-check"></i><b>14.4.3</b> Uniform distribution</a></li>
<li class="chapter" data-level="14.4.4" data-path="Chapter_14.html"><a href="Chapter_14.html#normal-distribution"><i class="fa fa-check"></i><b>14.4.4</b> Normal distribution</a></li>
</ul></li>
<li class="chapter" data-level="14.5" data-path="Chapter_14.html"><a href="Chapter_14.html#summary-2"><i class="fa fa-check"></i><b>14.5</b> Summary</a></li>
</ul></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> The Central Limit Theorem (CLT)</a>
<ul>
<li class="chapter" data-level="15.1" data-path="Chapter_15.html"><a href="Chapter_15.html#the-distribution-of-means-is-normal"><i class="fa fa-check"></i><b>15.1</b> The distribution of means is normal</a></li>
<li class="chapter" data-level="15.2" data-path="Chapter_15.html"><a href="Chapter_15.html#probability-and-z-scores"><i class="fa fa-check"></i><b>15.2</b> Probability and z-scores</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> <em>Practical</em>. Probability and simulation</a>
<ul>
<li class="chapter" data-level="16.1" data-path="Chapter_16.html"><a href="Chapter_16.html#probabilities-from-a-dataset"><i class="fa fa-check"></i><b>16.1</b> Probabilities from a dataset</a></li>
<li class="chapter" data-level="16.2" data-path="Chapter_16.html"><a href="Chapter_16.html#probabilities-from-a-normal-distribution"><i class="fa fa-check"></i><b>16.2</b> Probabilities from a normal distribution</a></li>
<li class="chapter" data-level="16.3" data-path="Chapter_16.html"><a href="Chapter_16.html#central-limit-theorem"><i class="fa fa-check"></i><b>16.3</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>Week 5 Overview</a></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> Confidence intervals (CIs)</a>
<ul>
<li class="chapter" data-level="17.1" data-path="Chapter_17.html"><a href="Chapter_17.html#normal-distribution-cis"><i class="fa fa-check"></i><b>17.1</b> Normal distribution CIs</a></li>
<li class="chapter" data-level="17.2" data-path="Chapter_17.html"><a href="Chapter_17.html#binomial-distribution-cis"><i class="fa fa-check"></i><b>17.2</b> Binomial distribution CIs</a></li>
</ul></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> The t-interval</a></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> <em>Practical</em>. z- and t- intervals</a>
<ul>
<li class="chapter" data-level="19.1" data-path="Chapter_19.html"><a href="Chapter_19.html#confidence-intervals-with-distraction"><i class="fa fa-check"></i><b>19.1</b> Confidence intervals with distrACTION</a></li>
<li class="chapter" data-level="19.2" data-path="Chapter_19.html"><a href="Chapter_19.html#confidence-intervals-from-z--and-t-scores"><i class="fa fa-check"></i><b>19.2</b> Confidence intervals from z- and t-scores</a></li>
<li class="chapter" data-level="19.3" data-path="Chapter_19.html"><a href="Chapter_19.html#confidence-intervals-for-different-sample-sizes-t--and-z-"><i class="fa fa-check"></i><b>19.3</b> Confidence intervals for different sample sizes (t- and z-)</a></li>
<li class="chapter" data-level="19.4" data-path="Chapter_19.html"><a href="Chapter_19.html#proportion-confidence-intervals"><i class="fa fa-check"></i><b>19.4</b> Proportion confidence intervals</a></li>
<li class="chapter" data-level="19.5" data-path="Chapter_19.html"><a href="Chapter_19.html#another-proportion-confidence-interval"><i class="fa fa-check"></i><b>19.5</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>Week 6 Overview</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> What is hypothesis testing?</a>
<ul>
<li class="chapter" data-level="20.1" data-path="Chapter_20.html"><a href="Chapter_20.html#how-ridiculous-is-our-hypothesis"><i class="fa fa-check"></i><b>20.1</b> How ridiculous is our hypothesis?</a></li>
<li class="chapter" data-level="20.2" data-path="Chapter_20.html"><a href="Chapter_20.html#statistical-hypothesis-testing"><i class="fa fa-check"></i><b>20.2</b> Statistical hypothesis testing</a></li>
<li class="chapter" data-level="20.3" data-path="Chapter_20.html"><a href="Chapter_20.html#p-values-false-positives-and-power"><i class="fa fa-check"></i><b>20.3</b> P-values, false positives, and power</a></li>
</ul></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> The t-test</a>
<ul>
<li class="chapter" data-level="21.1" data-path="Chapter_21.html"><a href="Chapter_21.html#one-sample-t-test"><i class="fa fa-check"></i><b>21.1</b> One sample t-test</a></li>
<li class="chapter" data-level="21.2" data-path="Chapter_21.html"><a href="Chapter_21.html#independent-samples-t-test"><i class="fa fa-check"></i><b>21.2</b> Independent samples t-test</a></li>
<li class="chapter" data-level="21.3" data-path="Chapter_21.html"><a href="Chapter_21.html#paired-sample-t-test"><i class="fa fa-check"></i><b>21.3</b> Paired sample t-test</a></li>
<li class="chapter" data-level="21.4" data-path="Chapter_21.html"><a href="Chapter_21.html#assumptions-of-t-tests"><i class="fa fa-check"></i><b>21.4</b> Assumptions of t-tests</a></li>
<li class="chapter" data-level="21.5" data-path="Chapter_21.html"><a href="Chapter_21.html#non-parametric-alternatives"><i class="fa fa-check"></i><b>21.5</b> Non-parametric alternatives</a>
<ul>
<li class="chapter" data-level="21.5.1" data-path="Chapter_21.html"><a href="Chapter_21.html#wilcoxon-test"><i class="fa fa-check"></i><b>21.5.1</b> Wilcoxon test</a></li>
<li class="chapter" data-level="21.5.2" data-path="Chapter_21.html"><a href="Chapter_21.html#mann-whitney-u-test"><i class="fa fa-check"></i><b>21.5.2</b> Mann-Whitney U test</a></li>
</ul></li>
<li class="chapter" data-level="21.6" data-path="Chapter_21.html"><a href="Chapter_21.html#summary-3"><i class="fa fa-check"></i><b>21.6</b> Summary</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> <em>Practical</em>. Hypothesis testing and t-tests</a>
<ul>
<li class="chapter" data-level="22.1" data-path="Chapter_22.html"><a href="Chapter_22.html#exercise-on-a-simple-one-sample-t-test"><i class="fa fa-check"></i><b>22.1</b> Exercise on a simple one sample t-test</a></li>
<li class="chapter" data-level="22.2" data-path="Chapter_22.html"><a href="Chapter_22.html#exercise-on-a-paired-t-test"><i class="fa fa-check"></i><b>22.2</b> Exercise on a paired t-test</a></li>
<li class="chapter" data-level="22.3" data-path="Chapter_22.html"><a href="Chapter_22.html#wilcoxon-test-1"><i class="fa fa-check"></i><b>22.3</b> Wilcoxon test</a></li>
<li class="chapter" data-level="22.4" data-path="Chapter_22.html"><a href="Chapter_22.html#independent-samples-t-test-1"><i class="fa fa-check"></i><b>22.4</b> Independent samples t-test</a></li>
<li class="chapter" data-level="22.5" data-path="Chapter_22.html"><a href="Chapter_22.html#mann-whitney-u-test-1"><i class="fa fa-check"></i><b>22.5</b> Mann-Whitney U Test</a></li>
</ul></li>
<li class="part"><span><b>VII Review of parts I-V</b></span></li>
<li class="chapter" data-level="" data-path="Week7.html"><a href="Week7.html"><i class="fa fa-check"></i>Week 7 Overview (Reading week)</a></li>
<li class="part"><span><b>VIII 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>Week 8 Overview</a></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> Analysis of variance</a>
<ul>
<li class="chapter" data-level="23.1" data-path="Chapter_23.html"><a href="Chapter_23.html#the-f-distribution"><i class="fa fa-check"></i><b>23.1</b> The F-distribution</a></li>
<li class="chapter" data-level="23.2" data-path="Chapter_23.html"><a href="Chapter_23.html#one-way-anova"><i class="fa fa-check"></i><b>23.2</b> One-way ANOVA</a>
<ul>
<li class="chapter" data-level="23.2.1" data-path="Chapter_23.html"><a href="Chapter_23.html#anova-mean-variance-among-groups"><i class="fa fa-check"></i><b>23.2.1</b> ANOVA mean variance among groups</a></li>
<li class="chapter" data-level="23.2.2" data-path="Chapter_23.html"><a href="Chapter_23.html#anova-mean-variance-within-groups"><i class="fa fa-check"></i><b>23.2.2</b> ANOVA mean variance within groups</a></li>
<li class="chapter" data-level="23.2.3" data-path="Chapter_23.html"><a href="Chapter_23.html#anova-f-statistic-calculation"><i class="fa fa-check"></i><b>23.2.3</b> ANOVA F statistic calculation</a></li>
</ul></li>
<li class="chapter" data-level="23.3" data-path="Chapter_23.html"><a href="Chapter_23.html#assumptions-of-anova"><i class="fa fa-check"></i><b>23.3</b> Assumptions of ANOVA</a></li>
</ul></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> Multiple comparisons</a></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> Kruskall-Wallis H test</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> Two-way ANOVA</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> <em>Practical</em>. ANOVA and associated tests</a>
<ul>
<li class="chapter" data-level="27.1" data-path="Chapter_27.html"><a href="Chapter_27.html#one-way-anova-site"><i class="fa fa-check"></i><b>27.1</b> One-way ANOVA (site)</a></li>
<li class="chapter" data-level="27.2" data-path="Chapter_27.html"><a href="Chapter_27.html#one-way-anova-profile"><i class="fa fa-check"></i><b>27.2</b> One-way ANOVA (profile)</a></li>
<li class="chapter" data-level="27.3" data-path="Chapter_27.html"><a href="Chapter_27.html#multiple-comparisons"><i class="fa fa-check"></i><b>27.3</b> Multiple comparisons</a></li>
<li class="chapter" data-level="27.4" data-path="Chapter_27.html"><a href="Chapter_27.html#kruskall-wallis-h-test"><i class="fa fa-check"></i><b>27.4</b> Kruskall-Wallis H test</a></li>
<li class="chapter" data-level="27.5" data-path="Chapter_27.html"><a href="Chapter_27.html#two-way-anova"><i class="fa fa-check"></i><b>27.5</b> Two-way ANOVA</a></li>
</ul></li>
<li class="part"><span><b>IX 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>Week 9 Overview</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> Frequency and count data</a>
<ul>
<li class="chapter" data-level="28.1" data-path="Chapter_28.html"><a href="Chapter_28.html#the-chi-square-distribution"><i class="fa fa-check"></i><b>28.1</b> The Chi-square distribution</a></li>
<li class="chapter" data-level="28.2" data-path="Chapter_28.html"><a href="Chapter_28.html#chi-squared-goodness-of-fit"><i class="fa fa-check"></i><b>28.2</b> Chi-squared goodness of fit</a></li>
<li class="chapter" data-level="28.3" data-path="Chapter_28.html"><a href="Chapter_28.html#chi-squared-test-of-association"><i class="fa fa-check"></i><b>28.3</b> Chi-squared test of association</a></li>
</ul></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> Correlation</a>
<ul>
<li class="chapter" data-level="29.1" data-path="Chapter_29.html"><a href="Chapter_29.html#scatterplots"><i class="fa fa-check"></i><b>29.1</b> Scatterplots</a></li>
<li class="chapter" data-level="29.2" data-path="Chapter_29.html"><a href="Chapter_29.html#the-correlation-coefficient"><i class="fa fa-check"></i><b>29.2</b> The correlation coefficient</a>
<ul>
<li class="chapter" data-level="29.2.1" data-path="Chapter_29.html"><a href="Chapter_29.html#pearson-product-moment-correlation-coefficient"><i class="fa fa-check"></i><b>29.2.1</b> Pearson product moment correlation coefficient</a></li>
<li class="chapter" data-level="29.2.2" data-path="Chapter_29.html"><a href="Chapter_29.html#spearman-rank-correlation-coefficient"><i class="fa fa-check"></i><b>29.2.2</b> Spearman rank correlation coefficient</a></li>
</ul></li>
<li class="chapter" data-level="29.3" data-path="Chapter_29.html"><a href="Chapter_29.html#correlation-hypothesis-testing"><i class="fa fa-check"></i><b>29.3</b> Correlation hypothesis testing</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> <em>Practical</em>. Analysis of counts and correlations</a>
<ul>
<li class="chapter" data-level="30.1" data-path="Chapter_30.html"><a href="Chapter_30.html#survival-goodness-of-fit"><i class="fa fa-check"></i><b>30.1</b> Survival goodness of fit</a></li>
<li class="chapter" data-level="30.2" data-path="Chapter_30.html"><a href="Chapter_30.html#colony-goodness-of-fit"><i class="fa fa-check"></i><b>30.2</b> Colony goodness of fit</a></li>
<li class="chapter" data-level="30.3" data-path="Chapter_30.html"><a href="Chapter_30.html#chi-square-test-of-association"><i class="fa fa-check"></i><b>30.3</b> Chi-Square test of association</a></li>
<li class="chapter" data-level="30.4" data-path="Chapter_30.html"><a href="Chapter_30.html#pearson-product-moment-correlation-test"><i class="fa fa-check"></i><b>30.4</b> Pearson product moment correlation test</a></li>
<li class="chapter" data-level="30.5" data-path="Chapter_30.html"><a href="Chapter_30.html#spearman-rank-correlation-test"><i class="fa fa-check"></i><b>30.5</b> Spearman rank correlation test</a></li>
<li class="chapter" data-level="30.6" data-path="Chapter_30.html"><a href="Chapter_30.html#untidy-goodness-of-fit"><i class="fa fa-check"></i><b>30.6</b> Untidy goodness of fit</a></li>
</ul></li>
<li class="part"><span><b>X Linear Regression</b></span></li>
<li class="chapter" data-level="" data-path="Week10.html"><a href="Week10.html"><i class="fa fa-check"></i>Week 10 Overview</a></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> Simple linear regression</a>
<ul>
<li class="chapter" data-level="31.1" data-path="Chapter_31.html"><a href="Chapter_31.html#visual-interpretation-of-regression"><i class="fa fa-check"></i><b>31.1</b> Visual interpretation of regression</a></li>
<li class="chapter" data-level="31.2" data-path="Chapter_31.html"><a href="Chapter_31.html#intercepts-slopes-and-residuals"><i class="fa fa-check"></i><b>31.2</b> Intercepts, slopes, and residuals</a></li>
<li class="chapter" data-level="31.3" data-path="Chapter_31.html"><a href="Chapter_31.html#regression-coefficients"><i class="fa fa-check"></i><b>31.3</b> Regression coefficients</a></li>
<li class="chapter" data-level="31.4" data-path="Chapter_31.html"><a href="Chapter_31.html#regression-line-calculation"><i class="fa fa-check"></i><b>31.4</b> Regression line calculation</a></li>
<li class="chapter" data-level="31.5" data-path="Chapter_31.html"><a href="Chapter_31.html#coefficient-of-determination"><i class="fa fa-check"></i><b>31.5</b> Coefficient of determination</a></li>
<li class="chapter" data-level="31.6" data-path="Chapter_31.html"><a href="Chapter_31.html#regression-assumptions"><i class="fa fa-check"></i><b>31.6</b> Regression assumptions</a></li>
<li class="chapter" data-level="31.7" data-path="Chapter_31.html"><a href="Chapter_31.html#regression-hypothesis-testing"><i class="fa fa-check"></i><b>31.7</b> Regression hypothesis testing</a>
<ul>
<li class="chapter" data-level="31.7.1" data-path="Chapter_31.html"><a href="Chapter_31.html#overall-model-significance"><i class="fa fa-check"></i><b>31.7.1</b> Overall model significance</a></li>
<li class="chapter" data-level="31.7.2" data-path="Chapter_31.html"><a href="Chapter_31.html#significance-of-the-intercept"><i class="fa fa-check"></i><b>31.7.2</b> Significance of the intercept</a></li>
<li class="chapter" data-level="31.7.3" data-path="Chapter_31.html"><a href="Chapter_31.html#significance-of-the-slope"><i class="fa fa-check"></i><b>31.7.3</b> Significance of the slope</a></li>
<li class="chapter" data-level="31.7.4" data-path="Chapter_31.html"><a href="Chapter_31.html#simple-regression-output"><i class="fa fa-check"></i><b>31.7.4</b> Simple regression output</a></li>
</ul></li>
<li class="chapter" data-level="31.8" data-path="Chapter_31.html"><a href="Chapter_31.html#conclusion"><i class="fa fa-check"></i><b>31.8</b> Conclusion</a></li>
</ul></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> Multiple regression</a></li>
<li class="chapter" data-level="33" data-path="adjusted-coefficient-of-determination.html"><a href="adjusted-coefficient-of-determination.html"><i class="fa fa-check"></i><b>33</b> Adjusted coefficient of determination</a></li>
<li class="chapter" data-level="34" data-path="Chapter_33.html"><a href="Chapter_33.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_33.html"><a href="Chapter_33.html#regression-exercise-1"><i class="fa fa-check"></i><b>34.1</b> Regression Exercise 1</a></li>
<li class="chapter" data-level="34.2" data-path="Chapter_33.html"><a href="Chapter_33.html#regression-exercise-2"><i class="fa fa-check"></i><b>34.2</b> Regression Exercise 2</a></li>
<li class="chapter" data-level="34.3" data-path="Chapter_33.html"><a href="Chapter_33.html#regression-exercise-3"><i class="fa fa-check"></i><b>34.3</b> Regression Exercise 3</a></li>
<li class="chapter" data-level="34.4" data-path="Chapter_33.html"><a href="Chapter_33.html#regression-exercise-4"><i class="fa fa-check"></i><b>34.4</b> Regression Exercise 4</a></li>
</ul></li>
<li class="part"><span><b>XI Randomisation approaches</b></span></li>
<li class="chapter" data-level="" data-path="Week11.html"><a href="Week11.html"><i class="fa fa-check"></i>Week 11 Overview</a></li>
<li class="chapter" data-level="35" data-path="introduction-to-randomisation.html"><a href="introduction-to-randomisation.html"><i class="fa fa-check"></i><b>35</b> Introduction to randomisation</a></li>
<li class="chapter" data-level="36" data-path="assumptions-of-randomisation.html"><a href="assumptions-of-randomisation.html"><i class="fa fa-check"></i><b>36</b> Assumptions of randomisation</a></li>
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<li class="chapter" data-level="39" data-path="practical.-using-r.html"><a href="practical.-using-r.html"><i class="fa fa-check"></i><b>39</b> <em>Practical</em>. Using R</a>
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<li class="chapter" data-level="39.2" data-path="practical.-using-r.html"><a href="practical.-using-r.html#r-exercise-2"><i class="fa fa-check"></i><b>39.2</b> R Exercise 2</a></li>
<li class="chapter" data-level="39.3" data-path="practical.-using-r.html"><a href="practical.-using-r.html#r-exercise-3"><i class="fa fa-check"></i><b>39.3</b> R Exercise 3</a></li>
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<li class="part"><span><b>XII Statistical Reporting</b></span></li>
<li class="chapter" data-level="" data-path="Week12.html"><a href="Week12.html"><i class="fa fa-check"></i>Week 12 Overview</a></li>
<li class="chapter" data-level="40" data-path="reporting-statistics.html"><a href="reporting-statistics.html"><i class="fa fa-check"></i><b>40</b> Reporting statistics</a></li>
<li class="chapter" data-level="41" data-path="more-introduction-to-r.html"><a href="more-introduction-to-r.html"><i class="fa fa-check"></i><b>41</b> More introduction to R</a></li>
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<li class="chapter" data-level="43" data-path="practical.-using-r-1.html"><a href="practical.-using-r-1.html"><i class="fa fa-check"></i><b>43</b> <em>Practical</em>. Using R</a>
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<li class="chapter" data-level="43.1" data-path="practical.-using-r-1.html"><a href="practical.-using-r-1.html#r-exercise-1-1"><i class="fa fa-check"></i><b>43.1</b> R Exercise 1</a></li>
<li class="chapter" data-level="43.2" data-path="practical.-using-r-1.html"><a href="practical.-using-r-1.html#r-exercise-2-1"><i class="fa fa-check"></i><b>43.2</b> R Exercise 2</a></li>
<li class="chapter" data-level="43.3" data-path="practical.-using-r-1.html"><a href="practical.-using-r-1.html#r-exercise-3-1"><i class="fa fa-check"></i><b>43.3</b> R Exercise 3</a></li>
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<li class="part"><span><b>XIII Review of parts (VII-XII)</b></span></li>
<li class="chapter" data-level="" data-path="Week13.html"><a href="Week13.html"><i class="fa fa-check"></i>Module summary</a></li>
<li class="appendix"><span><b>Appendix</b></span></li>
<li class="chapter" data-level="A" data-path="appendexA_CMS.html"><a href="appendexA_CMS.html"><i class="fa fa-check"></i><b>A</b> Common Marking Scheme</a></li>
<li class="chapter" data-level="B" data-path="uncertainty_derivation.html"><a href="uncertainty_derivation.html"><i class="fa fa-check"></i><b>B</b> Uncertainty derivation</a></li>
<li class="chapter" data-level="C" data-path="appendixC_tables.html"><a href="appendixC_tables.html"><i class="fa fa-check"></i><b>C</b> Statistical tables</a>
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<li class="chapter" data-level="C.1" data-path="appendixC_tables.html"><a href="appendixC_tables.html#wilcoxon-signed-rank-critical-values"><i class="fa fa-check"></i><b>C.1</b> Wilcoxon signed rank critical values</a></li>
<li class="chapter" data-level="C.2" data-path="appendixC_tables.html"><a href="appendixC_tables.html#mann-whitney-u-critical-values"><i class="fa fa-check"></i><b>C.2</b> Mann-Whitney U critical values</a></li>
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<h1><span class="header-section-number">Chapter 33</span> Adjusted coefficient of determination<a href="adjusted-coefficient-of-determination.html#adjusted-coefficient-of-determination" class="anchor-section" aria-label="Anchor link to header"></a></h1>
<p>Recall from <a href="Chapter_31.html#coefficient-of-determination">Chapter 31.5</a> that the coefficient of determination (<span class="math inline">\(R^{2}\)</span>) tells us how much of the total variation in the dependent variable is explained by the regression model.
This was fine for a simple linear regression, but with the addition of new independent variables, the proportion of the variance in y explained by our model is expected to increase even if the new independent variables are not very good predictors.
This is because the amount of variation explained by our model can only increase if we add new independent variables.
In other words, any new independent variable that we choose to add to the model cannot explain a negative amount of variation; that does not make any sense!
The absolute <em>worst</em> that an independent variable can do is explain zero variation.
And even if the independent variable is just a set of random numbers, it will likely explain <em>some</em> of the variation in the dependent variable just by chance.
Hence, even if newly added independent variables are bad predictors, they might still improve the goodness of fit of our model by chance.
To help account for this spurious improvement of fit, we can use an adjusted R squared (<span class="math inline">\(R^{2}_{adj}\)</span>).
The <span class="math inline">\(R^{2}_{adj}\)</span> takes into account the <span class="math inline">\(R^{2}\)</span>, the sample size (<span class="math inline">\(N\)</span>), and the number of independent variables (<span class="math inline">\(k\)</span>),</p>
<p><span class="math display">\[R^{2}_{adj} = 1 - \left(1 - R^{2}\right)\left(\frac{N - 1}{N - k - 1}\right).\]</span></p>
<p>As <span class="math inline">\(k\)</span> increases, the fraction above <span class="math inline">\((N-1)/(N-k-1)\)</span> gets bigger.
And as this fraction gets bigger, we are subtracting a bigger value from 1, so <span class="math inline">\(R^{2}\)</span> decreases.
Consequently, more independent variables (k) cause a decrease in the adjusted R-squared value.
This attempts to account for the inevitable tendency of <span class="math inline">\(R^{2}\)</span> to increase with <span class="math inline">\(k\)</span>.</p>
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