Multiple regression with dichotomous variable, and continuous regressor on which the groups differ
Dear list,
A couple of years ago Tom Nichols posted to the list a most informative document ( https://www.jiscmail.ac.uk/cgi-bin/wa.exe?A2=SPM;PPOtEw;20080318094435%2B0000) illustrating many common ANCOVA/Multiple regression models. Analyzing my data I have come across the need for one further design (an extension of Tom’s last model (ANCOVA with interaction)) – but I‘d like to check if my interpretations of coefficients and contrasts are accurate. I was wondering if you could spare a few minutes to read through and help me pick any errors. I have used the format Tom had used as I found it to offer great clarity. One essential difference from Tom’s last model is that I have centered the behavioural variable (C) within subgroup, because there are mean differences in C between genotype groups A and B. Would this be correct/necessary? If there are any other design implementations I would be all ears!
I am interested in the increase in response with increasing C, but wish to check if there is C x Group; if interactions were not significant, I suppose I should use a simpler model, with a single C column which would still contain values centred within subgroup?
Thank you all for your help!
Yannis
MULTIPLE REGRESSION, INCLUDING 2 GROUPS, 1 BEHAVIOURAL RESPONSE VARIABLE AND 1 COVARIATE OF NO INTEREST
GROUPS OF SUBJECTS: GENOTYPE A (N=4) AND GENOTYPE B (N=5), ONE CONTINUOUS BEHAVIOURAL VARIABLE: C, centered within subgroup
COVARIATE OF NO INTEREST: AGE, centered for total group
Design Matrix Parameterization
A B C*A C*B AGE
1 0 -20 0 -4
1 0 -5 0 6
1 0 0 0 -3
1 0 25 0 0
0 1 0 -20 -5
0 1 0 -5 4
0 1 0 0 -4
0 1 0 10 2
0 1 0 15 4
Coefficient interpretation
Beta 1: Expected response for Group A for C=Group_A_Average(C) and Age= TotalGroup_Average(Age), while accounting for the Group_A_specific linear effect of C and TotalGroup_specific linear effect of Age
Beta 2: Expected response for Group B for C=Group_B_Average(C) and Age= TotalGroup_Average(Age), while accounting for the Group_B_specific linear effect of C and TotalGroup_specific linear effect of Age
Beta 3: Expected change in response with increase of 1 unit of C for Group A, for Age= TotalGroup_Average(Age), while accounting for the TotalGroup_specific linear effect of Age
Beta 4: Expected change in response with increase of 1 unit of C for Group B, for Age= TotalGroup_Average(Age), while accounting for the TotalGroup_specific linear effect of Age
Beta 5: : Expected change in response with increase of 1 year of Age for TotalGroup,for C = 0 [??]
Contrast Interpretation
1 [T] [0 0 1 0 0] Increase in response in Group A with increasing C.
2 [T] [0 0 0 -1 0] Decrease in response in Group B with increasing C.
3 [T] [0 0 1 -1 0] Groups differ in C effects; Group A slope > Group B slope
4 [F] [0 0 -1 1 0] Groups differ in C effects; Group A slope > Group B slope OR Group B slope > Group A slope
5 [T] [0 0 1 1 0] Average increase in response in Groups A and B with increasing C.
6 [F] [0 0 -1 -1 0] Average decrease in response in Groups A and B with increasing C.
7 [T] [1 0 0 0 0] Average Group A response at C=Group_A_Average(C) and Age=TotalGroupAverage(Age), while accounting for Group_A_specific linear effect of C and Group_A-specific linear effect of C
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Yannis Paloyelis, BSc (Hons) MSc MSc
PhD Student
Office C3.18 (P080)
MRC Social Genetic and
Developmental Psychiatry Centre
Institute of Psychiatry,
King's College London
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