Chi squared
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<section xml:id="app-Chi-squared-table" xmlns:xi="http://www.w3.org/2001/XInclude">
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<title><m>\chi^2</m> Critical Values</title>
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<table xml:id="table-Chi-squared">
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<title><m>\chi^2</m> Critical Values</title>
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<tabular halign="center">
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<row bottom="minor" header="yes">
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<cell>Degrees of Freedom</cell>
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<cell>Critical value for <m>p = 0.05</m></cell>
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</row>
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<row>
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<cell><m>1</m></cell>
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<cell><m>3.841</m></cell>
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</row>
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<row>
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<cell><m>2</m></cell>
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<cell><m>5.991</m></cell>
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</row>
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<row>
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<cell><m>3</m></cell>
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<cell><m>7.815</m></cell>
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</row>
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<row>
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<cell><m>4</m></cell>
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<cell><m>9.488</m></cell>
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</row>
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<row>
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<cell><m>5</m></cell>
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<cell><m>11.070</m></cell>
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</row>
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<row>
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<cell><m>6</m></cell>
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<cell><m>12.592</m></cell>
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</row>
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<row>
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<cell><m>7</m></cell>
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<cell><m>14.067</m></cell>
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</row>
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<row>
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<cell><m>8</m></cell>
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<cell><m>15.507</m></cell>
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</row>
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<row>
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<cell><m>9</m></cell>
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<cell><m>16.919</m></cell>
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</row>
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<row>
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<cell><m>10</m></cell>
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<cell><m>18.307</m></cell>
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</row>
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</tabular>
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</table>
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</section>
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@@ -33,6 +33,12 @@
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<xi:include href="./app-Phi-table.ptx" />
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</appendix>
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<appendix>
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<title><m>\chi^2</m> Critical Values Table</title>
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<xi:include href="./app-Chi-squared-table.ptx" />
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</appendix>
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<colophon>
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<p>
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This book was authored in <pretext />.
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+218
-2
@@ -2,9 +2,225 @@
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<title><m>\chi^2</m> Test</title>
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<p>
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Text of section.
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In our previous hypothesis tests, we've asked about the correct value of some distribution parameter.
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But we might also be interested in whether a particular type of distribution fits the data in the first place.
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In this situation, we can try to measure the extent to which the observed data differs from what would be expected under a particular distribution.
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</p>
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<definition xml:id="def-chi-squared">
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<statement>
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<p>
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Suppose <m>O_1, \dotsc, O_n</m> are observed data with expected values <m>E_1, \dotsc, E_n</m>.
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The <term><m>\chi^2</m> statistic</term> is:
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<md>
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<mrow> \chi^2 = \sum_{i=1}^n \frac{(O_i - E_i)^2}{E_i}. </mrow>
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</md>
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</p>
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</statement>
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</definition>
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<p>
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The expression <m>O_i - E_i</m> directly measures the difference between an observation and its expected value.
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However, if we simply summed up these differences, then one observation which is higher than its expected value and another that's lower would have differences that cancel out.
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Squaring the term ensures that any difference contributes positively when summed together.
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And dividing by <m>E_i</m> ensures that the differences are expressed relative to the size of the expected value.
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A difference of <m>0.1</m> might sound like a small difference, but it would have a different interpretation if the expected value is <m>100</m> than if the expected value is <m>0.01</m>.
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</p>
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<p>
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So the overall number <m>\chi^2</m> is designed to be a measurement of how much the observed data differs from expectation.
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It seems like <m>\chi^2</m> could be higher or lower simply as the result of the number of terms involved in the summation.
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How much difference is enough to reject the distribution which generated those expected values? It seems that it should depend on that number of terms.
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</p>
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<definition xml:id="def-degrees-of-freedom-1">
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<statement>
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<p>
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If the data collected fall into <m>n</m> categories, then we will say that there are <m>n - 1</m> <term>degrees of freedom</term>.
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</p>
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</statement>
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</definition>
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<p>
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Suppose our data comes from polling <m>100</m> respondents and arranging their responses into <m>n</m> categories.
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Then, if we know how many people fall into categories <m>1, 2, \dotsc, n-1</m>, we will automatically know how many people fall into the final category, simply because the total number of respondents is already known.
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</p>
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<p>
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For each different number of degrees of freedom, there will be a different critical value of the <m>\chi^2</m> statistic, above which we'll say that the data differs sufficiently from expectation for us to reject the distribution being tested.
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A table of critical <m>\chi^2</m> values is provided in <xref ref="app-Chi-squared-table"/>.
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</p>
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<example>
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<statement>
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<p>
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Suppose <m>100</m> people are surveyed about their favorite color among four choices.
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The results are below.
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Is this data consistent with the hypothesis that each of the four colors is equally likely to be a favorite?
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</p>
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<tabular halign="center">
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<row bottom="minor" header="yes">
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<cell>Color</cell>
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<cell>Chosen as Favorite</cell>
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</row>
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<row>
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<cell>Blue</cell>
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<cell><m>37</m></cell>
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</row>
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<row>
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<cell>Green</cell>
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<cell><m>26</m></cell>
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</row>
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<row>
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<cell>Purple</cell>
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<cell><m>15</m></cell>
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</row>
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<row>
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<cell>Red</cell>
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<cell><m>22</m></cell>
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</row>
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</tabular>
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<p>
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In this case, if each color is equally likely to be a favorite, then we would expect <m>25</m> respondents to choose each color.
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So the <m>\chi^2</m> statistic is:
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<md>
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<mrow> \chi^2 = \frac{(37 - 25)^2}{25} + \frac{(26 - 25)^2}{25} + \frac{(15 - 25)^2}{25} + \frac{(22 - 25)^2}{25} = 10.16.</mrow>
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</md>
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Since the data fall into four categories, the number of degrees of freedom is <m>4 - 1 = 3</m>.
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From the table in <xref ref="table-Chi-squared"/>, the critical value is <m>7.815</m>.
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Since <m>10.16 \gt 7.815</m>, our collected data is sufficiently different to reject the hypothesis of equal color preferences.
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</p>
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</statement>
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</example>
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<p>
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In the previous example, we're testing an <term>extrinsic hypothesis</term>, i.e., a hypothesis made entirely without reference to the data.
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Knowing that we planned to poll <m>100</m> people, we could determine our expected values for this hypothesis before even conducting the survey.
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However, sometimes we need to use the collected data to estimate the value(s) of one or more parameters of a distribution.
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In such a situation, we say the hypothesis is <term>intrinsic</term>, and we adjust our calculation of degrees of freedom.
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</p>
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<definition xml:id="def-degrees-of-freedom-2">
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<statement>
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<p>
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Suppose it takes <m>m</m> measurements to fully specify the collected data, and suppose there are <m>k</m> parameters that are estimated using the data.
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Then the number of <term>degrees of freedom</term> is <m>m - k</m>
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</p>
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</statement>
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</definition>
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<example>
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<statement>
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<p>
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Is the data below consistent with a geometric distribution?
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</p>
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<tabular halign="center">
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<row bottom="minor" header="yes">
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<cell>Number</cell>
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<cell>Observation</cell>
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</row>
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<row>
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<cell><m>1</m></cell>
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<cell><m>59</m></cell>
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</row>
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<row>
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<cell><m>2</m></cell>
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<cell><m>21</m></cell>
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</row>
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<row>
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<cell><m>3</m></cell>
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<cell><m>11</m></cell>
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</row>
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<row>
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<cell><m>4</m></cell>
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<cell><m>8</m></cell>
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</row>
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<row>
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<cell><m>5</m></cell>
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<cell><m>1</m></cell>
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</row>
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</tabular>
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<p>
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In order to compare this data to a geometric distribution, we would need to know the value of the parameter <m>\theta</m> for the distribution.
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Based on the collected data, we could provide a maximum likelihood estimate:
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<md>
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<mrow> \widehat{\theta} \amp = \frac{59 + 21 + 11 + 8 + 1}{1 \cdot 59 + 2\cdot 21 + 3\cdot 11 + 4\cdot 8 + 5\cdot 1} </mrow>
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<mrow> \amp = \frac{100}{171} </mrow>
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<mrow> \amp \approx 0.58. </mrow>
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</md>
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We can use <xref ref="fact-geometric-probability-formula"/> with the parameter value <m>0.58</m> to generate expected values for our data.
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For example:
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<md>
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<mrow> g(2; 0.58) \amp = (1 - 0.58)^{1}(0.58) = 0.2436, </mrow>
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</md>
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so we would expect <m>0.2436 \times 100 = 24.36</m> observations for the number <m>2</m>.
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The rest of the expected values are calculated similarly:
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</p>
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<tabular halign="center">
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<row bottom="minor" header="yes">
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<cell>Number</cell>
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<cell>Observation</cell>
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<cell>Expected</cell>
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</row>
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<row>
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<cell><m>1</m></cell>
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<cell><m>59</m></cell>
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<cell><m>58</m></cell>
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</row>
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<row>
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<cell><m>2</m></cell>
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<cell><m>21</m></cell>
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<cell><m>24.36</m></cell>
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</row>
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<row>
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<cell><m>3</m></cell>
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<cell><m>11</m></cell>
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<cell><m>10.23</m></cell>
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</row>
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<row>
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<cell><m>4</m></cell>
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<cell><m>8</m></cell>
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<cell><m>4.30</m></cell>
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</row>
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<row>
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<cell><m>5</m></cell>
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<cell><m>1</m></cell>
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<cell><m>1.80</m></cell>
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</row>
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</tabular>
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<p>
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Now we can calculate the <m>\chi^2</m> statistic:
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<md>
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<mrow> \chi^2 \amp = \frac{(59 - 58)^2}{58} + \dotsb + \frac{(1 - 1.8)^2}{1.8} \approx 4.08. </mrow>
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</md>
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With <m>5</m> categories, it takes <m>4</m> measurements to fully specify the data.
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Then, we used the data itself to estimate the value of one parameter, <m>\widehat{\theta}</m>.
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So the number of degrees of freedom is <m>4 - 1 = 3</m>, with critical <m>\chi^2</m> value <m>7.815</m>.
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Since <m>4.08 \lt 7.815</m>, our data is not sufficiently different from expected to reject the hypothesis of a geometric distribution.
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</p>
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</statement>
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</example>
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<exercises xml:id="exercises-Chi-Squared">
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<exercise>
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<statement>
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@@ -101,4 +317,4 @@
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</task>
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</exercise>
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</exercises>
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</section>
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</section>
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