I have the following equations:
Yij = a + b1Ai+b2Dj
Y1 = 3830 + 26.5*A1 – 65*D1 (p- value b1 = 0.000; p-value b2 = 0.183)
Y2 = 5131 + 22.6*A2 – 125*D2 (p-value b1 = 0.000; p-value b2 = 0.004)
Y3 = 2631 + 18.2*A3 – 44.4*D3 (p-value b1 = 0.001; p-value b2 = 0.064)
Y4 = -2654 + 36*A4 + 31*D4 (p-value b1 = 0.000; p-value b2 = 0.266)
Question 1: How can I compare the slopes of the regression equations and obtain post hoc separation? I am particularly interested in the slope of A.
Question 2: The p-values of the coefficient of D most times show non significance and at other times it is significant. This implies that a minimal model without D should suffice in such cases. Therefore, how can I compare regressions with count of b values being different?
You should fit a model that includes all these 4 groups. This model can then estimate the differences between the coefficients (slopes) between groups.
It is cumbersome to write the explicit regression equations, but it is simple to give the "model formulas". Your regression models would be Y ~ A + D (Y being modelled as a linear combination of the factors/predictors A and D). You need a model that allows the coefficients (betas) to be different for 4 groups. Using "Group" as an additional categorical predictor in your model, this is modelled as the interaction of A and D and the group: Y ~ (A+D)*Group.
This model will contain coefficients that estimate the differences in the slopes over A (and D) between the groups. Usually, the factor "Group" is dummy-coded, giving three columns in the design matrix that refer to the difference of three groups against one reference group. If you want the differences between any two groups you can either refit the model with definig a different reference group or you can use some software that gives you these contrasts directly from your model.
You should fit a model that includes all these 4 groups. This model can then estimate the differences between the coefficients (slopes) between groups.
It is cumbersome to write the explicit regression equations, but it is simple to give the "model formulas". Your regression models would be Y ~ A + D (Y being modelled as a linear combination of the factors/predictors A and D). You need a model that allows the coefficients (betas) to be different for 4 groups. Using "Group" as an additional categorical predictor in your model, this is modelled as the interaction of A and D and the group: Y ~ (A+D)*Group.
This model will contain coefficients that estimate the differences in the slopes over A (and D) between the groups. Usually, the factor "Group" is dummy-coded, giving three columns in the design matrix that refer to the difference of three groups against one reference group. If you want the differences between any two groups you can either refit the model with definig a different reference group or you can use some software that gives you these contrasts directly from your model.
Yes, you can depend on p-value which is smallest(p-value go to zero) is more effect.
aslo answer of Mr. Jochen is very good you can consider new factore(dummay var.) as Group and ANOVA giv a good results.
with best regards
Thank you Jochen for the insight. It worked quite well. However, I think I ran into another situation. If I keep re-leveling the reference group, b values keep changing. Which b value should I report, is it the initial b value before applying dummy variables?
After relevelling, the b-values have different meanings. Report those that have a meaning relevant to your research question.
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