ANCOVA using SPSS 'Regression' & 'General Linear Model'



kxQ Models: Main Effects & Interaction Models Using Regression

The purpose of this study was to examine the relationship of practice and exam performance. Two aspects of practice were selected for study, the difficulty of the practice and the number of practices completed. Practice difficulty was a 3-condition variable - practice problems were either about the same difficulty as the exam problems (=1), they were easier than the exam problems (=2), or they were more difficult than the exam problems (=3). Different sections of the course were randomly assigned to receive the three difficulty levels. Students were permitted to complete as many practice problems as they liked, receiving very complete feedback after each problem. The dependent variable was performance on an examination.

| |[pic] |

|Here are the group means for number of practices and exam performance. | |

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|There is a practice difficulty group difference for number of practices – those| |

|in the easier condition and the harder condition completed fewer practices | |

|than those in the same condition. | |

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|There is also a practice difficulty group difference for test performance. | |

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|Notice the confounding – different practice difficulty groups have different | |

|number of practices, making it difficult to know whether to attribute the test | |

|performance differences to the practice difficulty, practice amount or their | |

|combined effect. | |

Data Preparation

Here’s the SPSS syntax code to dummy code the binary grouping variable, to center the quantitative variable (using mean overall mean for that variables from above) and to compute the interaction term.

▪ Dummy coding follows the GLM convention – the group with the highest original code as the control group

▪ Centering of quantitative variables simplifies interpretation of the regression weights

▪ Interactions are “non-additive combinations” -- meaning products of the related main effects

|* pract_dc1 compares same=1=>0 with easier = 2 => 1. |IF statements to dummy-code the group variable: |

|if (practgrp = 1) pract_dc1 = 0. |same is going to be the comparison group, so it is coded “0” for both dummy |

|if (practgrp = 2) pract_dc1 = 1. |codes |

|if (practgrp = 3) pract_dc1 = 0. |dc1 is going to compare easier with same, so easier is coded “1” as the |

| |target group & same is coded “0” (harder is also coded “0”) |

|*pract_dc2 compare same=1=>0 with harder=3=>1. |dc2 is going to compare harder with same, so harder is coded as “1” as the |

|if (practgrp = 1) pract_dc2 = 0. |target group & same is codec “0” (easier is also coded “0”) |

|if (practgrp = 2) pract_dc2 = 0. | |

|if (practgrp = 3) pract_dc2 = 1. |Centering the covariate requires subtracting the mean from each person’s |

| |number of practices score |

|compute numpract_cen = numpract - 5.792. | |

| |The product of each of the dummy codes with the centered quantitative are |

|compute grp_pract_int1 = pract_dc1 * numpract_cen. |the interaction terms |

|compute grp_pract_int2 = pract_dc2 * numpract_cen. | |

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|exe. | |

|*main effects model. |*full model with interaction. |

|REGRESSION |REGRESSION |

|/STATISTICS COEFF R ANOVA |/STATISTICS COEFF R ANOVA |

|/DEPENDENT testperf |/DEPENDENT testperf |

|/METHOD=ENTER numpract_cen |/METHOD=ENTER numpract_cen |

|. pract_dc1 pract_dc2 |. pract_dc1 pract_dc2 |

| |. grp_pract_int1 grp_pract_int2 |

Main effects model or ANCOVA assuming regression slope homogeneity (no interaction)

|[pic] |[pic] |

|[pic] | |

| |The model “works” significantly better than chance. |

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| |This model accounts for about 32% of the variance in |

| |the performance scores. |

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Interpreting the regression weights

|constant |The expected value of testperf when the value of all predictors = 0 |

| |The expected value of testperf for those in the same condition and who have 5.792 practices |

| |Those in easier condition who had 5.792 practices scored 65.259% on the exam |

|numpract_cen |The direction and extent of the expected change in testperf for a 1-unit increase in this predictor, holding the value of the other predictor |

| |constant at 0 |

| |The expected change in testperf for each additional practice, for those in the same condition |

| |For each additional practice, those in the same condition are expected to increase 2.689 % on testperf – this effect is significant |

| |Note: because there is no interaction term in the model, and, so, the slopes of the three group’s lines must be the same, this is also the |

| |slope of the relationship between practice and performance for the easier and harder conditions |

|pract_dc1 |The direction and extent of the pairwise group difference, holding the other predictor constant at 0. |

| |The same vs. easier group difference controlling the number of practices at 0 (the mean after centering) |

|compares same & easier|Those in the same group outperformed those in the easier condition by 13.178%, when holding for the number of practices at 5.792 – this effect |

| |is significant. |

| |So, the corrected mean for the same condition when practice is controlled at 5.792 is 65.259% (constant) and the corrected mean difference |

| |between the groups is 13.178%, (dc1 regression weight), so the corrected mean for the easier condition is 52.081% |

| |Notice that this corrected pairwise group difference is different from the uncorrected pairwise group difference (67.5 vs. 50.625). |

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|pract_dc2 |The direction and extent of the pairwise group difference, holding the other predictor constant at 0. |

| |The same vs. harder group difference controlling the number of practices at 0 (the mean after centering) |

|compares same & harder|Those in the harder group outperformed those in the same condition by 1.775%, when holding for the number of practices at 5.792 – this effect |

| |is not significant. |

| |So, the corrected mean for the samer condition when practice is controlled at 5.792is 65.259% (constant) and the corrected mean difference |

| |between the groups is 1.775%, (dc2 regression weight), so the corrected mean for the easier condition is 67.034% |

| |Notice that this corrected pairwise group difference is in the opposite direction of the uncorrected pairwise group difference – though neither|

| |pairwise comparison is significantly different. |

Obtaining & Interpreting the Plot of the Model

| [pic] | |

| |Using the “kxQ” tab of the Excel file… |

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| |Fill in the values from the analyses. |

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| |Be sure you label the groups correctly & get the right regression |

| |weight into the correct spot. |

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| |Put “0” in for the interaction regression weights. |

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| |The weights for the simple regression line for each group are |

| |automatically calculated. |

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| |Notice the slope of the simple regression line for each group is the |

| |same – with no interaction term in the model we are forcing a |

| |homogeneity of regression slope assumption onto the model. |

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| |The only difference between the two simple regression lines is their |

| |height, which is given by the regression weights for the dummy codes. |

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| |The plotting points for the graph are automatically computed. Both |

| |the raw and the centered x-axis values are shown. |

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| |Because there is no interaction term, the main effects are “safe”. |

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| |So, the pattern of pairwise group difference when controlling at the |

| |mean number of practices generalizes into a pattern of pairwise group |

| |differnces at all amounts of practice. |

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Full model, including the interaction

|[pic] |[pic] |

|[pic] | |

| |The model “works” significantly better than chance. |

| | |

| |This model accounts for about 76% of the variance in |

| |the performance scores. |

| | |

Interpreting the regression weights

|constant |The expected value of testperf when the value of all predictors (practice difficulty, number of practices and interaction) = 0 |

| |The expected value of testperf for those in the easier condition and who have 5.792 practices |

| |Those in easier condition who had 5.938 practices scored about 64.756% on the exam |

|numpract_cen |The direction and extent of the expected change in testperf for a 1-unit increase in this predictor, holding the value of the other |

| |predictors (practice & interaction) constant at 0 |

| |The expected change in testperf as the number of practice changes for those in the same condition |

| |For each additional practice, those in the easier condition are expected to decrease .963 % on testperf – this effect is significant |

| |Note: because there is an interaction term in the model, the slopes of the three group’s lines may be different – check the |

| |interaction to evaluate this. |

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|pract_dc1 |The direction and extent of this pairwise group difference, holding the other predictors (number of practices and the interaction |

| |constant at 0. |

|compares same & easier |The same vs. easier group difference controlling the number of practices and the interaction at 0 (their mean after centering) |

| |Those in the same group outperformed those in the easier condition by 16.038%, when holding for the number of practices at 5.792 – |

| |this effect is significant. |

| |So, the corrected mean for the same condition when practice is controlled at 5.792 is 64.756% (constant) and corrected mean difference|

| |between the groups is 16.038%, (pairwise regression weight), so the corrected mean for the easier condition is 48.718% |

| |Notice that this corrected pairwise group difference is different from the uncorrected pairwise difference between the groups. |

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|pract_dc2 |The direction and extent of this pairwise group difference, holding the other predictors (number of practices and the interaction |

| |constant at 0. |

|compares same & harder |The same vs. harder group difference controlling the number of practices and the interaction at 0 (their mean after centering) |

| |Those in the harder group outperformed those in the same condition by 3.732%, when holding for the number of practices at 5.792 – this|

| |effect is not significant. |

| |So, the corrected mean for the easier condition when practice is controlled at 5.792 is 64.756% (constant) and corrected mean |

| |difference between the groups is 3.732% (pairwise regression weight), so the corrected mean for the same condition is 68.488% |

| |Notice that this corrected pairwise group difference is different from the uncorrected pairwise difference between the groups. |

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|grp_pract_int1 |The direction and extent of the difference in the testperf-numpract slope for these two groups. |

| |The direction and extent of change in the practice difficulty pairwise group difference for each 1-unit increase in number of |

| |practices |

|compare slopes of same |How the practice group difficulty changes as the number of practices changes |

|& easier groups |For each additional practice, the difference between the similar difficulty practice group and the easier practice group increases by |

| |-6.182% – this effect is significant. |

| |So, for those in the same practice group performance increases by 3.392 for each practice, whereas for those in the easier practice |

| |group, performance decreases by 2.89 (3.292 - 6.182). |

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|grp_pract_int2 |The direction and extent of the difference in the testperf-numpract slope for these two groups. |

| |The direction and extent of change in the practice difficulty pairwise group difference for each 1-unit increase in number of |

|compare slopes of same |practices |

|& harder groups |How the practice group difficulty changes as the number of practices changes |

| |For each additional practice, the difference between the similar difficulty practice group and the harder practice group increases by |

| |4.38 – this effect is significant. |

| |So, for those in the similar difficulty practice group performance decreases by 3.292 for each practice, whereas for those in the |

| |harder difficulty practice group, performance increases by 7.672 (3.292 + 4.38). |

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Obtaining & Interpreting the Plot of the Model

| [pic] | |

| |Using the “kxQ” tab of the Excel file… |

| | |

| | |

| |Fill in the values from the analyses. |

| | |

| | |

| |Be sure you label the groups correctly! |

| | |

| | |

| | |

| | |

| | |

| | |

| |The weights for the simple regression line for each group are |

| |automatically calculated. |

| | |

| |Notice the slope of the simple regression line for each group is different|

| |– with an interaction term in the model we are testing homogeneity of |

| |regression slope assumption onto the model – and rejecting it! |

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| | |

| |The plotting points for the graph are automatically computed. Both the |

| |raw and the centered x-axis values are shown. |

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| |We can see the pattern of the interaction: |

| |Performance increases with practice for the same difficulty group but |

| |decreases (non- significantly) for those in the easier practice group. |

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| |From this interaction pattern we can see that the the main effect of |

| |practice would have to be misleading. |

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| |Similarly, we can see that the group effect varies from one direction to |

| |the other, rendering the main effect for group misleading. |

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