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



2xQ GLM/Regression Model: Coding & Centering

For models that include interactions, regression weights describe simple effects (not main effects) and differences between simple effects. The dummy coding of qualitative variables determines which group’s criterion-predictor simple regression line is tested. The centering of quantitative variables determines at what value of the quantitative variable the group mean differences are “controlled” and compared.

Dummy Coding Binary Variables Here is the original coding we used ( I’ve changed the variable names sp that they carry specific information about what group plays what role (comparison = 0 vs. target = 1)

|*same = 1 easier = 2. |pg_s1e0 – practice group ( same is target & easy is comparison |

|if (practgrp = 1) pg_s1e0 = 1. |will test the simple effect difference between the same and easier conditions when numpract_meancen = 0 |

|if (practgrp = 2) pg_s1e0 = 0. |which is a simple effect group comparison when numpract = 5.938 |

| | |

| |numpract_meancen |

|compute |will test the slope of the testperf-practice regression line when pg_s1e0 = 0 |

|numpract_meancen |a test of the slope of the testperf-practice regression line for the easier group |

|= numpract - 5.938. | |

| |grps1e0_practmean_int |

| |will test the direction and extent of the difference in slope of the testperf-practice regression line between the |

|compute |groups |

|grps1e0_practmean_int |a test of how the slope of the testperf-practice regression line for the same (=1) group differs from that of the |

|= pg_s1e0 * numpract_meancen. |easier (=0) group. |

| | |

|exe.. | |

We got…

|[pic] |[pic] |

|[pic] | |

|[pic] | |

constant ( mean of easier group (=0) when numpract_meancen = 0 (pract = 5.938) is 56.832%

pg_s1e0 ( the same group performs 8.406% better than the easier group when numpract_meancen = 0 (pract = 5.938)

numpract_meancen ( for the easier group each additional practice decreases performance by -.971%

grps1e0_practmean_int

• For each additional practice, the difference between the similar difficulty practice group and the easier practice group becomes 4.264% larger – notice that the direction “flips” at about 4 practices

• For those in the easier practice group performance decreases by .971% for each practice, whereas for those in the similar difficulty practice group, performance increases by 3.292% (-.971 + 4.263).

Here is a different dummy coding of the 2-group practice difficulty variable.

|*same = 1 easier = 2. |pg_s0e1 – practice group ( easy is target & same is comparison |

|if (practgrp = 1) pg_s0e1 = 0. |will test the simple effect difference between the same and easier conditions when numpract_meancen = 0 |

|if (practgrp = 2) pg_s0e1 = 1. |which is a simple effect group comparison when numpract = 5.938 |

| | |

|compute |numpract_meancen |

|numpract_meancen |will test the slope of the testperf-practice regression line when pract_dc1 – 0 |

|= numpract - 5.938. |a test of the slope of the testperf-practice regression line for the easier group |

| | |

|compute |interaction |

|grps0e1_practmean_int |will test the direction and extent of the difference in slope of the testperf-practice regression line between the |

|= pg_s0e1 * numpract_meancen. |groups |

| |a test of how the slope of the testperf-practice regression line for the same (=1) group differs from that of the |

|exe. |easier (=0) group. |

|[pic] |[pic] |

|[pic] | |

|[pic] | |

constant ( mean of same group when numpract_meancen = 0 (pract = 5.938) is 65.238%

pg_s0e1 ( the easier group performs 8.406% poorer than the easier group when numpract_meancen = 0 (pract = 5.938)

numpract_meancen ( for the same group each additional practice increases performance by 3.292%

grps0e1_practmean_int

• For each additional practice, the difference between the similar difficulty practice group and the easier practice group becomes 4.264% smaller – notice that the direction “flips” at about 4 practices

• For those in the same practice group performance increases by 3.292% for each practice, whereas for those in the easier practice group, performance increases by -.971 (3.292 - 4.263).

What did we replicate & what did we “learn new” by re-coding the binary variable??

Notice that the simple regression lines and the pictures from the two codings show “the same model”

pg_s1e0 ( numpract_meancen tests H0: b=0 for the testpert – numpract regression slope for the Easy group (=0)

pg_s0e1 ( numpract_meancen tests H0: b=0 for the testpert – numpract regression slope for the Same group (=0)

pg_s1e0 ( both test the simple effect mean difference between groups when numpract_meancen = 0

pg_s0e1 ( - have opposite signs

grps1e0_practmean_int ( both test the interaction

grps0e1_practmean_int ( - have opposite signs

Centering Quantitative Variables Here is the original coding we used – with “upgraded” the variable names…

|*same = 1 easier = 2. |pg_s1e0 – practice group ( same is target – easy is comparison |

|if (practgrp = 1) pg_s1e0 = 1. |will test the simple effect difference between the same and easier conditions when num_pract_cen = 0 |

|if (practgrp = 2) pg_s1e0 = 0. |which is a simple effect group comparison when numpract = 5.938 |

| | |

| |numpract_meancen |

| |will test the slope of the testperf-practice regression line when pg_s1e0 = 0 |

|compute numpract_ |a test of the slope of the testperf-practice regression line for the easier group |

|meancen | |

|= numpract - 5.938. |grps1e0_practmean_int |

| |will test the direction and extent of the difference in slope of the testperf-practice regression line between |

| |the groups |

|compute |a test of how the slope of the testperf-practice regression line for the same (=1) group differs from that of the|

|grps1e0_practmean_int |easier (=0) group. |

|= pg_s1e0 * numpract_meancen. | |

| | |

|exe.. | |

We got…

|[pic] |[pic] |

|[pic] | |

|[pic] | |

constant ( mean of easier group when numpract_meancen = 0 (pract = 5.938) is 56.832%

pg_s1e0 ( the same group performs 8.406% better than the easier group when numpract_meancen = 0 (pract = 5.938)

numpract_meancen ( for the easier group each additional practice decreases performance by -.971%

grps1e0_practmean_int

• For each additional practice, the difference between the similar difficulty practice group and the easier practice group becomes 4.264% larger – notice that the direction “flips” at about 4 practices

• For those in the easier practice group performance decreases by .971% for each practice, whereas for those in the similar difficulty practice group, performance increases by 3.292% (-.971 + 4.263).

So, the only value of practice for which we have a direct test of the group mean difference is the mean, 5.938 (centered to 0). Since the pattern of the interaction is that the group difference increases as the number of practices increases, we can infer that there is also a group difference for every larger value of practice. But what about smaller value? What about the interesting reversal at small value of practice?

You can test the group difference at any value of the quantitative variable, simply by centering the quantitative variable at that value and computing the associated interaction term.

Not many folks are likely to practice “.612” times. But what if a person only practices one time? Is there a same-easy practice difficulty effect when there is a single practice? What we want to test is the simple effect of group at a practice value of 1.

Remember that when a model includes an interaction term, the other terms are simple effects, not main effects. Each tells the effect of that variable (group difference or regression line slope) when the other variable is held constant at “0”. So, at whatever value we “center” the other variable, making it “0,” is the value at which the group comparison will be controlled and made.

When we re-center a variable

• The regression weight for the re-centered variable will not change (re-centering is an additive linear transformation, which will not change the slope of the regression line between that variable and the criterion for the comparison group in the dummy code

• The interaction will not change, because the re-centering does not change the slope difference between the two dummy-coded groups

• The regression weight of the dummy-coded variable will change ( telling the group mean difference at the centered value of the quantitative variable

• The constant will change ( telling the mean of the comparison group at the centered value of the quantitative variable

|Here is the code to center the number of |[pic] |

|practices at 1.0 and compute the associated | |

|interaction term. | |

| | |

|if (practgrp = 1) pg_s0e1 = 0. | |

|if (practgrp = 2) pg_s0e1 = 1. | |

| | |

|compute | |

|numpract_1cen | |

|= numpract - 1.0. | |

| | |

|compute | |

|grps1e0_pract1cen_int | |

|= pg_s1e0 * numpract_1cen. | |

| | |

|exe. | |

| |[pic] |

| |[pic] |

As expected, the regression weights for the number of practices centered at 1.0 and the interaction are the same as in the earlier analysis (that mean centered).

• The constant tells us that when practice is held constant at 1 (0 after centering), the Easy group had a mean of 61.626%.

• The regression weight for the dummy code tells us that the Easy group did 12.644% poorer than the Same group, when number of practices is held constant at 1 (0 after centering). At 1 practice, the Easy group had an average performance of 61.626% and the Same group had an average performance of 48.98 ( 61.626 + (-12.646) ).

This simple effect mean difference is not significant. Huh?

When holding the number of practices constant at 5.938 (mean centering) the group difference of 8.406% was statistically significant (p = .020). But this “larger” simple effect of 12.646 is not significant (p = .092)??

Notice that the Std. Error is much smaller when we mean centered the quantitative variable (3.392) than when we centered it at 1.0 (7.239). This larger error term for 1-centering than for mean-centering led to a smaller t-value, even with the larger simple effect mean difference.

In general, there is more error in the estimation of y’ from a regression line the farther away from the mean of the predictor is the point of estimation. This is why specific tests of group differences at meaningful values of the quantitative variable are so important.

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