There was a pretty good thread on this general topic, recently, with
subject "Data Screening". You'd do well to look that thread up. I'll
give citations, which are all from that thread; the comments are too
long to post here.
I'll take the assumptions in a convenient order, rather than the order
you've given them.
>4) Normality
The assumption is normal distribution of the residuals, not of the DV
or any IVs. Hector Maletta posted an extensive discussion; date-time is
Wed, 28 Sep 2005 22:08:00 -0300.
>1) the assumption of linearity
Theory may indicate a non-linear relationship, in which case it's
proper to transform the variables so that the theoretically expected
relationship is linear.
If theory is lacking, you usually have to make the assumption of
linearity and live with it, because the data will not show any
deviation from it. That means, however, that within the accuracy of
your measurements, any deviations don't matter.
You can test linearity directly by including higher-order terms,
typically quadratic to start with, in your model, and testing for their
significance as a group. However,
- The quadratic terms can be so highly correlated with the linear terms
that the resulting model can't be estimated. There are formal
procedures to handle this, but if you're using only quadratic terms,
it's usually enough to change the measurement origin of the IVs so
their means are near 0, certainly less than 1 SD.
- Adding quadratic terms, with product terms, adds a lot of degrees of
freedom to the model: n(n+1)/2, if you have n IVs in the model. Very
often you won't have enough data for a model that size.
- Unless you have a pretty high R-square in the linear model (sorry, I
can't give you numbers), you have little hope of 'seeing' non-linear
effects.
See also my posting Wed, 28 Sep 2005 20:17:13 -0400, in the cited
thread. (The discussion of non-linearity starts a ways down the post.)
>2) the assumptions of independence
You said you "have ensured that each measurement is independent by
randomization". Can you say more what your study is, and how you drew
the samples? And, given that you drew randomly from your available
data, could you instead have used all your available data?
It's also assumed that the residuals are statistically uncorrelated
with the IVs (not necessary independent). Nothing much you can do about
that, except adopt the convention that the portion that's correlated
with (explainable by) the IVs is part of the DV, not the residual.
If time is an IV, successive residuals can fail to be independent
because the process hasn't had enough time to change between
measurements. Statistics such as the Durbin-Watson are used for this.
Analogous problems can theoretically arise with very closely-spaced
measurements of other IVs, but I don't think this is considered a
common problem in practice.
>3) the assumption of constant variance [of the residuals]
I.e., homoscedasticity. Sometimes theory will indicate higher
measurement errors in different parts of the DV range, which can
sometimes be addressed by transformations. Hector discussed this in his
post I cited above, dated Wed, 28 Sep 2005 22:08:00 -0300.
If you have an ANOVA problem, i.e. multiple measures for the same
values of the DVs, you can check. See CELLPLOTS command in MANOVA
(Advanced Models module). If you have a variable that you suspect is a
measure of the residual variance, you can use WLS (Regression Models
module).
See also Hector's discussion of residuals, twice cited above.
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