You know what happens when you assume? I remember learning this little jewel of
wisdom in a junior high school shop class. Our shop teacher wrote the word
assume on the chalkboard. a, s, s, u, m, e (and yes we had chalkboards back then).
Drawing a line under the first three letters of the word assume, he told us
that when you assume, you make an ass out of you and me. Just like in shop class, in
statistics you should never assume that the tests you are using are appropriate
until you check your assumptions. So why do assumptions matter? Let's say that you
call me on my cell phone, and I don't answer. You conclude that whenever you
call I must be picking up my phone, I see that it's you calling, and I press that
little F U button at the top of the phone that sends your call to voicemail.
Perhaps you conclude that I'm mad at you and I'm ignoring your calls. But what is
your assumption? You are assuming that I had my phone with me, and for me,
personally, that is never a safe assumption. So next time we meet and you
ask me why I was ignoring your phone calls I explain that I left my phone
on the charger at home again, and with me that's probably pretty likely what
happened. But what can we learn about assumptions? If your assumption is wrong,
it prevents you from looking at the world accurately. If your assumptions are
incorrect, what is going to be true about your conclusions based on those
assumptions? Your conclusions are going to be incorrect as well. Now parametric
statistics are built on certain assumptions about the data, and if those
assumptions are violated, if the assumptions are incorrect, what is going
to be true about your conclusions based on those assumptions? Your conclusions
are going to be incorrect as well. You're more likely to be wrong. In fact, Collier
et al. wrote ". . .all models are limited by the validity
of the assumptions on which they ride." so we need to know about the assumptions of
the statistical tests that we are using. Parametric statistics require that the
population have certain characteristics. Parametric statistics based on the
normal distribution assume that the data are the interval or ratio level,
independent scores - no confounds or covariates - the assumption of normality,
and the assumption of homogeneity of variance. So let's look at each one of
those. First, the assumption of interval or ratio data. This is an absolute
necessity for your dependent variable scores. You can't do parametric
statistics with nominal DV data. You can do other kinds of tests, but not
parametric tests. The assumption of interval or ratio data is easily
verified. Look at your data. Have lots of ones and zeros? You have categorical data
Lots of numbers within a range? Now you can do parametric statistics. The
assumption of independent scores - each individual in the sample should be
independent of the others. You do not have before and after scores or people
in one group influencing people in another group. Now if this assumption is
violated, you can still use parametric statistics, but you need to switch to a
repeated measures design. Or, if you have potential confounds or covariates, you
have to control for that in your research methodology as well. Satisfying
the assumption of independent scores is done during research construction. You
should know what kind of tests you were going to use even before you begin your
data collection. The assumption of normality - this is a tricky one - a very
common misconception is that the data in your sample must be normally distributed
in order to use parametric statistics, or you may hear that the population from
which the data are drawn must be normally distributed.
Actually, the assumption of normality is that the distribution of the means of
the samples drawn from the population are normally distributed. Watch my video
on the central limit theorem for a fuller explanation of this assumption of
normality. But regardless, you can examine the distribution of your sample scores
using graphs like the QQ plot or the histogram, or you can use parametric
tests for normality, such as the kolmogorov-smirnov test. Again,
understanding the central limit theorem will help you see that parametric tests
can be used even when the sample data you're testing are not normally
distributed, because the distribution of sample means will be normally
distributed, even if the parent population is not. The assumption of
homogeneity of variance - the variability in each group should be approximately
equal. This assumption is really only necessary when you're comparing groups,
not with repeated measures. The assumption of homogeneity of variance
can be easily tested with Levene's test. With multiple repeated measures tests
however, you should also check for sphericity (more about sphericity in a
lecture on repeated measures ANOVA). so now you know what you need to know to know to
check your assumptions, and I have some good news. Parametric statistics are
robust. That means that even if your sample data violate the assumption of
normality, if they're not normally distributed, you can still use
inferential procedures. Again the lecture on central limit theorem will help
explain why, or you can just trust me on this. However the more that your data
deviate from a normal distribution, the more error enters the tests, so you
should be particularly careful on interpretations, when the result of your
statistical test is right on that p equals .05 cutoff.
But then, what do you do if you can't use parametric tests, or if you don't want to,
or if your advisor tells you that you can't use parametric tests? Good news. You
have options. You can still use nonparametric statistics. Nonparametric
statistics are used when the assumptions are too badly violated to use parametric
statistics. Nonparametric statistics do not require the stringent assumptions
about the population. They can be used with nominal and ordinal data, and this
is their real strength. They can also be used with highly skewed distributions.
However, parametric statistics are more powerful than nonparametric, meaning that
you are more likely to find an effect that actually exists i. e. avoid a type 2
error, if you use parametric statistics. Also, these nonparametric statistics
cannot be used with complex research designs, and that can be pretty limiting.
And that is how you check your assumptions. So with that, you now know
what you need to know to make sure that when you're doing statistics, you don't
make an ass out of you and me.