# What does the Scale parameter mean in linear regression?

I have started to use the GENLIN procedure in SPSS more than any of the specific dialogues, but I don't understand the Scale parameter or why it has the effects it does on the regression results.

Here's an SPSS example: code block 1

Normally, if I wanted to see if set has an effect on the linear parameters $b_0$ and $b_1$, I'd could just do linear regression with the interaction term set1X: code block 2

The results are obvious: set affects slope but not the intercept of my model (as I'd expect since I generated these sample data that way). But if I use GENLIN instead of REGRESSION, my sig. values and 95%CIs for the parameters are different: code block 3

This Scale parameter (regressed by GENLIN as 1.127 with a SE of 0.3563) seems to make the difference. If I change the scale to PEARSON (for Pearson's chi sq.), DEVIANCE (?), or a constant like 1, I get all different answers.

Summary: what is Scale in Generalized Linear Models (GENLIN in SPSS) and how should I handle it? Why doesn't OLS Linear regression use such a parameter? How do I know how to assign scale?

## Code block 1

data list list /X set Y.
begin data.
1   1   7.0
2   1   7.8
3   1   12.4
4   1   14.8
5   1   19.0
6   1   22.7
7   1   24.4
8   1   25.5
9   1   29.5
10  1   31.0
1   2   7.9
2   2   12.7
3   2   14.3
4   2   20.1
5   2   20.8
6   2   26.5
7   2   30.9
8   2   35.8
9   2   38.0
10  2   43.7
end data.
dataset name exampleData WINDOW=front.
variable level X (scale) Y (scale) set (nominal).

compute set1 = (set=1).
compute set1X = set1*X.
EXECUTE.


## Code block 2

REGRESSION
/MISSING LISTWISE
/STATISTICS COEFF OUTS CI(95) R ANOVA CHANGE
/CRITERIA=PIN(.05) POUT(.10)
/NOORIGIN
/DEPENDENT Y
/METHOD=FORWARD X set1 set1X.


## Code block 3

* Generalized Linear Models.
GENLIN Y BY set (ORDER=ASCENDING) WITH X
/MODEL X set*X INTERCEPT=YES
/CRITERIA SCALE=MLE COVB=MODEL PCONVERGE=1E-006(ABSOLUTE) SINGULAR=1E-012 ANALYSISTYPE=3(WALD)
CILEVEL=95 CITYPE=WALD LIKELIHOOD=FULL
/MISSING CLASSMISSING=EXCLUDE
/PRINT CPS DESCRIPTIVES MODELINFO FIT SUMMARY SOLUTION.

• Scale parameter makes it possible to account for nonhomogeneity variances stats.stackexchange.com/q/48237/3277 – ttnphns May 18 '13 at 5:24
• Does this apply to models that don't use probit or logit? My example specifically uses two continuous variables. Does scale still just account for heteroscedasticity? – DocBuckets May 18 '13 at 16:36

You're missing "set" as an effect in GENLIN (in your MODEL statement) but you have it in REGRESSION. Try this:

GENLIN Y BY set (ORDER=ASCENDING) WITH X
/MODEL set X set*X INTERCEPT=YES DISTRIBUTION=NORMAL LINK=IDENTITY
/CRITERIA SCALE=MLE COVB=MODEL PCONVERGE=1E-006(ABSOLUTE) SINGULAR=1E-012
ANALYSISTYPE=3(WALD) CILEVEL=95 CITYPE=WALD LIKELIHOOD=FULL
/MISSING CLASSMISSING=EXCLUDE
/PRINT CPS DESCRIPTIVES MODELINFO FIT SUMMARY SOLUTION.


That will give you the results you're looking for.

The scale parameter is actually called "dispersion" in most generalized linear model theory papers or articles. See here for more information.