// Numbas version: finer_feedback_settings
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"tsqovern", "mu4", "mu1", "g", "r2", "mu2", "ss", "f1tsqovern", "r3", "f2ss", "mrs", "f1rss", "f2l1", "f2l2", "r1", "stovern", "t", "f1stovern", "f2t", "n1", "f2stovern", "n3", "n4"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"dec": {"type": "string", "language": "jme", "definition": "switch(vr>=12.88, 'Very strongly reject the null hypothesis ($p\\\\le 0.001$)',vr>=7.41,'Strongly reject the null hypothesis ($0.001 \\\\lt p \\\\le 0.01$)',vr>=4.12, 'Reject the null hypothesis ($0.01 \\\\lt p \\\\le 0.05$)',vr>=2.86,'Accept the null hypothesis ($0.05 \\\\lt p \\\\le 0.1$)','Accept the null hypothesis($p\\\\gt 0.1$)') ", "parameters": [["vr", "number"]]}, "pstdev": {"type": "number", "language": "jme", "definition": "sqrt(abs(l)/(abs(l)-1))*stdev(l)", "parameters": [["l", "list"]]}}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "1", "maxValue": "1", "marks": 0.25}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "f1btss-tol", "maxValue": "f1btss+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "f1btss-tol", "maxValue": "f1btss+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "f1vr-tol", "maxValue": "f1vr+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "1", "maxValue": "1", "marks": 0.25}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "f2btss-tol", "maxValue": "f2btss+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "f2btss-tol", "maxValue": "f2btss+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "f2vr-tol", "maxValue": "f2vr+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "1", "maxValue": "1", "marks": 0.25}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "interactionss-tol", "maxValue": "interactionss+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "interactionss-tol", "maxValue": "interactionss+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "ivr-tol", "maxValue": "ivr+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "36", "maxValue": "36", "marks": 0.25}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "rss-tol", "maxValue": "rss+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "mrs-tol", "maxValue": "mrs+tol", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "39", "maxValue": "39", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "precround(f1btss+f2btss+interactionss+rss,2)-tol", "maxValue": "precround(f1btss+f2btss+interactionss+rss,2)+tol", "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "m1", "maxValue": "m1", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "m2", "maxValue": "m2", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "m3", "maxValue": "m3", "marks": 0.5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "m4", "maxValue": "m4", "marks": 0.5}], "type": "gapfill", "prompt": "<p>Now complete the following two factor ANOVA table from this data. Input the $SS,\\;MS$ and $VR$ data to 2 decimal places.</p>\n<table border=\"1\">\n<tbody>\n<tr><th align=\"left\">Source</th><th>df</th><th>SS</th><th>MS</th><th>VR</th></tr>\n<tr><th align=\"left\">Personality</th>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td>[[3]]</td>\n</tr>\n<tr><th align=\"left\">Stimulus</th>\n<td>[[4]]</td>\n<td>[[5]]</td>\n<td>[[6]]</td>\n<td>[[7]]</td>\n</tr>\n<tr><th align=\"left\">P-S interaction</th>\n<td>[[8]]</td>\n<td>[[9]]</td>\n<td>[[10]]</td>\n<td>[[11]]</td>\n</tr>\n<tr><th align=\"left\">Residual</th>\n<td>[[12]]</td>\n<td>[[13]]</td>\n<td>[[14]]</td>\n<td>-</td>\n</tr>\n<tr><th align=\"left\">Total</th>\n<td>[[15]]</td>\n<td>[[16]]</td>\n<td>-</td>\n<td>-</td>\n</tr>\n</tbody>\n</table>\n<p><strong>The Calculations.</strong></p>\n<p>1. Residual Calculations. When you are calculating $TSS$ and $BTSS$ round them both to 2 decimal places, then calculate $RSS$ by taking away these rounded values. The $RMS$ value is then obtained by dividing this $RSS$ value by the residual degrees of freedom.</p>\n<p>2. For Personality, Stimulus and Interaction, calculate the estimations of their variances to 2 decimal places as well. These values go in the $MS$ column.</p>\n<p>3. The $VR$ values are obtained by dividing the first three values in the $MS$ column by the $RMS$ value. Enter the $VR$ values to 2 decimal places in the last column.</p>\n<p>&nbsp;</p>\n<p>Also input the mean values of the factors at their various levels, input your answers to 2 decimal places:</p>\n<p>&nbsp;</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td>&nbsp;</td>\n<td>$\\overline{x}_i$</td>\n</tr>\n<tr><th align=\"left\">Normal, Baseline</th>\n<td>[[17]]</td>\n</tr>\n<tr><th align=\"left\">Normal, Stress</th>\n<td>[[18]]</td>\n</tr>\n<tr><th align=\"left\">Antisocial, Baseline</th>\n<td>[[19]]</td>\n</tr>\n<tr><th align=\"left\">Antisocial, Stress</th>\n<td>[[20]]</td>\n</tr>\n</tbody>\n</table>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"layout": {"expression": ""}, "choices": ["GSR: no dependence upon personality.", "GSR:  no dependence upon stimulus.", "No interaction between the factors."], "matrix": "w", "type": "m_n_x", "maxAnswers": 0, "shuffleChoices": false, "marks": 0, "scripts": {}, "minMarks": 0, "minAnswers": 0, "maxMarks": 0, "shuffleAnswers": false, "showCorrectAnswer": true, "answers": ["Very strong rejection, $p \\le 0.001$.", "Strong rejection,  $0.001\\lt p \\le 0.01$.", "Rejection, $0.01\\lt p \\le 0.05$.", "Accept, but investigate, $0.05\\lt p \\le 0.1$.", "Accept, $p \\gt 0.1$."]}], "type": "gapfill", "prompt": "<p>Using the following p-values for the $F_{1,36}$ statistic find the appropriate significance levels for the factors as given by their $VR$ value and then comment on the null hypotheses for each factor and the interaction.</p>\n        <p>&nbsp;</p>\n        <table border=\"1\">\n        <tbody>\n        <tr>\n        <td>$10\\%$</td>\n        <td>$5\\%$</td>\n        <td>$1\\%$</td>\n        <td>$0.1\\%$</td>\n        </tr>\n        <tr>\n        <td>$2.86$</td>\n        <td>$4.12$</td>\n        <td>$7.41$</td>\n        <td>$12.88$</td>\n        </tr>\n        </tbody>\n        </table>\n        <p>&nbsp;</p>\n        <p>[[0]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>&nbsp;In a similar fashion to the calculation done in lectures, carry out a two factor ANOVA on&nbsp;the following set of data.</p>\n  <p>Individuals who are identified as having an antisocial personality disorder may also have reduced physiological responses to anxiety-producing stimuli.</p>\n  <p>One way of measuring this response is with &rdquo;galvanic skin response&rdquo; (GSR), a measurable&nbsp;reduction in the electrical resistance on the skin. [This is the basis of how a lie detector&nbsp;works.]</p>\n  <p>The following data represent the results of an experiment to compare the responses&nbsp;of normal and antisocial individuals in regular (baseline) and stress-provoking situations (low score reflects a more anxious individual):</p>\n  <table border=\"1\">\n  <tbody>\n  <tr><th rowspan=\"2\">Normal</th><th>Baseline</th>\n  <td>$\\var{r1[0]}$</td>\n  <td>$\\var{r1[1]}$</td>\n  <td>$\\var{r1[2]}$</td>\n  <td>$\\var{r1[3]}$</td>\n  <td>$\\var{r1[4]}$</td>\n  <td>$\\var{r1[5]}$</td>\n  <td>$\\var{r1[6]}$</td>\n  <td>$\\var{r1[7]}$</td>\n  <td>$\\var{r1[8]}$</td>\n  <td>$\\var{r1[9]}$</td>\n  </tr>\n  <tr><th>Stress</th>\n  <td>$\\var{r2[0]}$</td>\n  <td>$\\var{r2[1]}$</td>\n  <td>$\\var{r2[2]}$</td>\n  <td>$\\var{r2[3]}$</td>\n  <td>$\\var{r2[4]}$</td>\n  <td>$\\var{r2[5]}$</td>\n  <td>$\\var{r2[6]}$</td>\n  <td>$\\var{r2[7]}$</td>\n  <td>$\\var{r2[8]}$</td>\n  <td>$\\var{r2[9]}$</td>\n  </tr>\n  <tr><th rowspan=\"2\">Antisocial</th><th>Baseline</th>\n  <td>$\\var{r3[0]}$</td>\n  <td>$\\var{r3[1]}$</td>\n  <td>$\\var{r3[2]}$</td>\n  <td>$\\var{r3[3]}$</td>\n  <td>$\\var{r3[4]}$</td>\n  <td>$\\var{r3[5]}$</td>\n  <td>$\\var{r3[6]}$</td>\n  <td>$\\var{r3[7]}$</td>\n  <td>$\\var{r3[8]}$</td>\n  <td>$\\var{r3[9]}$</td>\n  </tr>\n  <tr><th>Stress</th>\n  <td>$\\var{r4[0]}$</td>\n  <td>$\\var{r4[1]}$</td>\n  <td>$\\var{r4[2]}$</td>\n  <td>$\\var{r4[3]}$</td>\n  <td>$\\var{r4[4]}$</td>\n  <td>$\\var{r4[5]}$</td>\n  <td>$\\var{r4[6]}$</td>\n  <td>$\\var{r4[7]}$</td>\n  <td>$\\var{r4[8]}$</td>\n  <td>$\\var{r4[9]}$</td>\n  </tr>\n  </tbody>\n  </table>\n  <p>&nbsp;The null hypotheses we are testing are:</p>\n  <p>1. The GSR does not depend upon the personality type.</p>\n  <p>2. The GSR does not depend upon the stimulus.</p>\n  <p>3. There is no interaction between personality and stimulus in determining the GSR.</p>\n  <p>&nbsp;</p>", "tags": ["ANOVA", "average", "checked2015", "data analysis", "degrees of freedom", "F-test", "factors", "hypothesis testing", "levels", "mean", "mean ", "p values", "PSY2010", "statistics", "stats", "two factor ANOVA", "variance"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>26/11/2012</strong></p>\n                                                                                    <p><strong><br /></strong>Added tags and description.</p>\n                                                                                    <p>Changed calculations for BTSS for the factors and added the interaction analysis.</p>\n                                                                                    <p>Uses the inbuilt ftest function from the stats extension to find p values.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Two factor ANOVA example</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>We have two factors: Personality and Stimulus.</p>\n    <p>Personality has two levels: Normal and Antisocial</p>\n    <p>Stimulus has two levels: Baseline and Stress.</p>\n    <p><strong>First Step:</strong></p>\n    <p>First regard the different treatment combinations as a set of independent samples and analyse&nbsp;as for a one-way analysis with unrelated measurements. From this analysis,&nbsp;we obtain the Total Sum of Squares, the Between Treatments Sum of Squares ($BTSS$) and Residual&nbsp;Sum of Squares ($RSS$). Note that the degrees of freedom for this step are 36=40-4 as there are 4 treatments.</p>\n    <p>You should obtain:</p>\n    <table border=\"1\">\n    <tbody>\n    <tr>\n    <td>&nbsp;</td>\n    <td>$\\overline{x}_i$</td>\n    <td>$T_i$</td>\n    <td>$\\sum x^2$</td>\n    <td>$n_i$</td>\n    </tr>\n    <tr><th align=\"left\">Normal, Baseline</th>\n    <td>$\\var{m1}$</td>\n    <td>$\\var{t[0]}$</td>\n    <td>$\\var{ssq[0]}$</td>\n    <td>10</td>\n    </tr>\n    <tr><th align=\"left\">Normal, Stress</th>\n    <td>$\\var{m2}$</td>\n    <td>$\\var{t[1]}$</td>\n    <td>$\\var{ssq[1]}$</td>\n    <td>10</td>\n    </tr>\n    <tr><th align=\"left\">Antisocial, Baseline</th>\n    <td>$\\var{m3}$</td>\n    <td>$\\var{t[2]}$</td>\n    <td>$\\var{ssq[2]}$</td>\n    <td>10</td>\n    </tr>\n    <tr><th align=\"left\">Antisocial, Stress</th>\n    <td>$\\var{m4}$</td>\n    <td>$\\var{t[3]}$</td>\n    <td>$\\var{ssq[3]}$</td>\n    <td>10</td>\n    </tr>\n    <tr>\n    <td>&nbsp;</td>\n    <td>&nbsp;</td>\n    <td>$G=\\;\\var{g}$</td>\n    <td>Sum of Squares=$\\var{ss}$</td>\n    <td>$N=40$</td>\n    </tr>\n    </tbody>\n    </table>\n    <p>From this we obtain:</p>\n    <p>\\[ BTSS =\\sum \\frac{T_i^2}{10}- \\frac{G^2}{40}=\\frac{\\var{t[0]}^2}{10}+\\frac{\\var{t[1]}^2}{10}+\\frac{\\var{t[2]}^2}{10}+\\frac{\\var{t[3]}^2}{10}-\\frac{\\var{g}^2}{40}=\\var{btss}\\]</p>\n    <p>\\[TSS = \\sum \\sum x^2- \\frac{G^2}{40}=\\var{ss}- \\frac{\\var{g}^2}{40}=\\var{tss}\\] both&nbsp;to 2 decimal places.</p>\n    <p>\\[RSS = TSS-BTSS=\\var{tss}-\\var{btss}=\\var{rss}\\]</p>\n    <p><strong>Second Step.</strong></p>\n    <p>Now ignore the Stimulus factor and calculate totals $T_i$ for each level of Personality. From these totals calculate&nbsp;a variance estimate for personality using the same method as before. The degrees of&nbsp;freedom will be one fewer than the levels of Stimulus and is therefore 1.</p>\n    <table border=\"1\">\n    <tbody>\n    <tr><th>Normal (Baseline and Stress)</th>\n    <td>$\\var{r1[0]}$</td>\n    <td>$\\var{r1[1]}$</td>\n    <td>$\\var{r1[2]}$</td>\n    <td>$\\var{r1[3]}$</td>\n    <td>$\\var{r1[4]}$</td>\n    <td>$\\var{r1[5]}$</td>\n    <td>$\\var{r1[6]}$</td>\n    <td>$\\var{r1[7]}$</td>\n    <td>$\\var{r1[8]}$</td>\n    <td>$\\var{r1[9]}$</td>\n    <td>$\\var{r2[0]}$</td>\n    <td>$\\var{r2[1]}$</td>\n    <td>$\\var{r2[2]}$</td>\n    <td>$\\var{r2[3]}$</td>\n    <td>$\\var{r2[4]}$</td>\n    <td>$\\var{r2[5]}$</td>\n    <td>$\\var{r2[6]}$</td>\n    <td>$\\var{r2[7]}$</td>\n    <td>$\\var{r2[8]}$</td>\n    <td>$\\var{r2[9]}$</td>\n    </tr>\n    <tr><th>Antisocial (Baseline and Stress)</th>\n    <td>$\\var{r3[0]}$</td>\n    <td>$\\var{r3[1]}$</td>\n    <td>$\\var{r3[2]}$</td>\n    <td>$\\var{r3[3]}$</td>\n    <td>$\\var{r3[4]}$</td>\n    <td>$\\var{r3[5]}$</td>\n    <td>$\\var{r3[6]}$</td>\n    <td>$\\var{r3[7]}$</td>\n    <td>$\\var{r3[8]}$</td>\n    <td>$\\var{r3[9]}$</td>\n    <td>$\\var{r3[0]}$</td>\n    <td>$\\var{r4[1]}$</td>\n    <td>$\\var{r4[2]}$</td>\n    <td>$\\var{r4[3]}$</td>\n    <td>$\\var{r4[4]}$</td>\n    <td>$\\var{r4[5]}$</td>\n    <td>$\\var{r4[6]}$</td>\n    <td>$\\var{r4[7]}$</td>\n    <td>$\\var{r4[8]}$</td>\n    <td>$\\var{r4[9]}$</td>\n    </tr>\n    </tbody>\n    </table>\n    <p>You should have the following data from this table:</p>\n    <p>&nbsp;</p>\n    <table border=\"1\">\n    <tbody>\n    <tr>\n    <td>&nbsp;</td>\n    <td>$T_i$</td>\n    <td>$\\sum x^2$</td>\n    <td>$n_i$&nbsp;(number of observations)</td>\n    </tr>\n    <tr><th align=\"left\">Normal, (Baseline and Stress)</th>\n    <td>$\\var{f1t[0]}$</td>\n    <td>$\\var{f1ssq[0]}$</td>\n    <td>20</td>\n    </tr>\n    <tr><th align=\"left\">Antisocial, (Baseline and Stress)</th>\n    <td>$\\var{f1t[1]}$</td>\n    <td>$\\var{f1ssq[1]}$</td>\n    <td>20</td>\n    </tr>\n    <tr>\n    <td>&nbsp;</td>\n    <td>$G=\\;\\var{g}$</td>\n    <td>Sum of Squares=$\\var{f1ss}$</td>\n    <td>$N=40$</td>\n    </tr>\n    </tbody>\n    </table>\n    <p>So we can calculate:</p>\n    <p>\\[\\mbox{Variance estimate for personality} =\\sum \\frac{T_i^2}{20}- \\frac{G^2}{40}=\\frac{\\var{f1t[0]}^2}{20}+\\frac{\\var{f1t[1]}^2}{20}-\\frac{\\var{g}^2}{40}=\\var{f1btss}\\]</p>\n    <p><strong>Third Step.</strong></p>\n    <p>Repeat step 2 with the factors switched, i.e. use the totals $T_i$ for the Stimulus factor levels ignoring Personality.This gives a Between Treatments Sum of Squares. Again the degrees of freedom will be 2-1=1.</p>\n    <table border=\"1\">\n    <tbody>\n    <tr><th>Baseline (Normal and Antisocial)</th>\n    <td>$\\var{r1[0]}$</td>\n    <td>$\\var{r1[1]}$</td>\n    <td>$\\var{r1[2]}$</td>\n    <td>$\\var{r1[3]}$</td>\n    <td>$\\var{r1[4]}$</td>\n    <td>$\\var{r1[5]}$</td>\n    <td>$\\var{r1[6]}$</td>\n    <td>$\\var{r1[7]}$</td>\n    <td>$\\var{r1[8]}$</td>\n    <td>$\\var{r1[9]}$</td>\n    <td>$\\var{r2[0]}$</td>\n    <td>$\\var{r3[1]}$</td>\n    <td>$\\var{r3[2]}$</td>\n    <td>$\\var{r3[3]}$</td>\n    <td>$\\var{r3[4]}$</td>\n    <td>$\\var{r3[5]}$</td>\n    <td>$\\var{r3[6]}$</td>\n    <td>$\\var{r3[7]}$</td>\n    <td>$\\var{r3[8]}$</td>\n    <td>$\\var{r3[9]}$</td>\n    </tr>\n    <tr><th>Stress (Normal and Antisocial)</th>\n    <td>$\\var{r2[0]}$</td>\n    <td>$\\var{r2[1]}$</td>\n    <td>$\\var{r2[2]}$</td>\n    <td>$\\var{r2[3]}$</td>\n    <td>$\\var{r2[4]}$</td>\n    <td>$\\var{r2[5]}$</td>\n    <td>$\\var{r2[6]}$</td>\n    <td>$\\var{r2[7]}$</td>\n    <td>$\\var{r2[8]}$</td>\n    <td>$\\var{r2[9]}$</td>\n    <td>$\\var{r4[0]}$</td>\n    <td>$\\var{r4[1]}$</td>\n    <td>$\\var{r4[2]}$</td>\n    <td>$\\var{r4[3]}$</td>\n    <td>$\\var{r4[4]}$</td>\n    <td>$\\var{r4[5]}$</td>\n    <td>$\\var{r4[6]}$</td>\n    <td>$\\var{r4[7]}$</td>\n    <td>$\\var{r4[8]}$</td>\n    <td>$\\var{r4[9]}$</td>\n    </tr>\n    </tbody>\n    </table>\n    <p>You should have the following data from this table:</p>\n    <p>&nbsp;</p>\n    <table border=\"1\">\n    <tbody>\n    <tr>\n    <td>&nbsp;</td>\n    <td>$T_i$</td>\n    <td>$\\sum x^2$</td>\n    <td>$n_i$ (number of observations)</td>\n    </tr>\n    <tr><th align=\"left\">Baseline (Normal and Antisocial)</th>\n    <td>$\\var{f2t[0]}$</td>\n    <td>$\\var{f2ssq[0]}$</td>\n    <td>20</td>\n    </tr>\n    <tr><th align=\"left\">Stress (Normal and Antisocial)</th>\n    <td>$\\var{f2t[1]}$</td>\n    <td>$\\var{f2ssq[1]}$</td>\n    <td>20</td>\n    </tr>\n    <tr>\n    <td>&nbsp;</td>\n    <td>$G=\\;\\var{g}$</td>\n    <td>Sum of Squares=$\\var{f2ss}$</td>\n    <td>$N=40$</td>\n    </tr>\n    </tbody>\n    </table>\n    <p>So we can calculate:</p>\n    <p>\\[ \\mbox{Variance estimate for stimulus} =\\sum \\frac{T_i^2}{20}- \\frac{G^2}{40}=\\frac{\\var{f2t[0]}^2}{20}+\\frac{\\var{f2t[1]}^2}{20}-\\frac{\\var{g}^2}{40}=\\var{f2btss}\\]</p>\n    <p><strong>Step 4.</strong></p>\n    <p>Now determine a Sum of Squares for Interaction by subtracting the sums of squares obtained&nbsp;for Personality (Step 2) and Stimulus (step 3) from the overall Between Treatments Sum of&nbsp;squares obtained in Step 1. The degrees of freedom is also obtained by subtraction and is 1.</p>\n    <p>This gives:&nbsp;</p>\n    <p>\\[\\mbox{Variance estimate for the interaction}= \\var{btss}-\\var{f1btss}-\\var{f2btss} = \\var{interactionss}\\]</p>\n    <p><strong>The Anova Table.</strong></p>\n    <p>We now have all the terms required to construct the ANOVA table and hence test the null&nbsp;hypothesis relating to each factor and to the interaction. Note that the $VR$ values are obtained by dividing the $RMS$ value into the $MS$ values for the factors and the interaction.</p>\n    <table border=\"1\">\n    <tbody>\n    <tr><th align=\"left\">Source</th><th>df</th><th>SS</th><th>MS</th><th>VR</th><th>Decision</th></tr>\n    <tr><th align=\"left\">Personality</th>\n    <td>1</td>\n    <td>$\\var{f1btss}$</td>\n    <td>$\\var{f1btss}$</td>\n    <td>$\\var{f1vr}$</td>\n    <td>{dec(f1vr)} that GSR is independent of personality.</td>\n    </tr>\n    <tr><th align=\"left\">Stimulus</th>\n    <td>1</td>\n    <td>$\\var{f2btss}$</td>\n    <td>$\\var{f2btss}$</td>\n    <td>$\\var{f2vr}$</td>\n    <td>{dec(f2vr)}&nbsp;that GSR is independent of the stimulus.</td>\n    </tr>\n    <tr><th align=\"left\">P-S interaction</th>\n    <td>1</td>\n    <td>&nbsp;$\\var{interactionss}$</td>\n    <td>&nbsp;&nbsp;$\\var{interactionss}$</td>\n    <td>&nbsp;$\\var{ivr}$</td>\n    <td>&nbsp;{dec(ivr)} that personality and stimulus are independent in terms of GSR.</td>\n    </tr>\n    <tr><th align=\"left\">Residual</th>\n    <td>36</td>\n    <td>$\\var{rss}$</td>\n    <td>$\\var{mrs}$</td>\n    <td>-</td>\n    <td>&nbsp;</td>\n    </tr>\n    <tr><th align=\"left\">Total</th>\n    <td>39</td>\n    <td>$\\var{precround(f1btss+f2btss+interactionss+rss,2)}$</td>\n    <td>-</td>\n    <td>-</td>\n    <td>&nbsp;</td>\n    </tr>\n    </tbody>\n    </table>"}]}], "contributors": [{"name": "", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/862/"}], "extensions": ["stats"], "custom_part_types": [], "resources": []}