Our research plans to use one-way ANOVA, but upon encountering the assumptions of it, we had to conduct the test for normality and homoscedasticity. We're going to compare 3 populations(Grade 10,11,12 students) with 2 dependent variables(parental perfectionism, career indecision) to be done separately.
I have decided upon using the Anderson-Darling Test, but my problem is, will I be testing each population for each dependent variable, or combine the three populations for each dependent variable and perform the Anderson-Darling test?
Solved – How to conduct test for normality before conducting an ANOVA
anovanormality-assumption
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Most statistics packages have ways of saving residuals from your model. Using GLM - UNIVARIATE in SPSS you can save residuals. This will add a variable to your data file representing the residual for each observation.
Once you have your residuals you can then examine them to see whether they are normally distributed, homoscedastic, and so on. For example, you could use a formal normality test on your residual variable or perhaps more appropriately, you could plot the residuals to check for any major departures from normality. If you want to examine homoscedasticity, you could get a plot that looked at the residuals by group.
For a basic between subjects factorial ANOVA, where homogeneity of variance holds, normality within cells means normality of residuals because your model in ANOVA is to predict group means. Thus, the residual is just the difference between group means and observed data.
Response to comments below:
- Residuals are defined relative to your model predictions. In this case your model predictions are your cell means. It is a more generalisable way of thinking about assumption testing if you focus on plotting the residuals rather than plotting individual cell means, even if in this particular case, they are basically the same. For example, if you add a covariate (ANCOVA), residuals would be more appropriate to examine than distributions within cells.
- For purposes of examining normality, standardised and unstandardised residuals will provide the same answer. Standardised residuals can be useful when you are trying to identify data that is poorly modelled by the data (i.e., an outlier).
- Homogeneity of variance and homoscedasticity mean the same thing as far as I'm aware. Once again, it is common to examine this assumption by comparing the variances across groups/cells. In your case, whether you calculate variance in residuals for each cell or based on the raw data in each cell, you will get the same values. However, you can also plot residuals on the y-axis and predicted values on the x-axis. This is a more generalisable approach as it is also applicable to other situations such as where you add covariates or you are doing multiple regression.
- A point was raised below that when you have heteroscedasticity (i.e., within cell variance varies between cells in the population) and normally distributed residuals within cells, the resulting distribution of all residuals would be non-normal. The result would be a mixture distribution of variables with mean of zero and different variances with proportions relative to cell sizes. The resulting distribution will have no zero skew, but would presumably have some amount of kurtosis. If you divide residuals by their corresponding within-cell standard deviation, then you could remove the effect heteroscedasticity; plotting the residuals that result would provide an overall test of whether residuals are normally distributed independent of any heteroscedasticity.
SPSS aside (I can't help you with that, sorry, I haven't used SPSS in decades), it's a relatively simple matter to use bootstrapping in an ANOVA, but before one even tries to do that it's important to consider what is being assumed and whether it makes sense with your variables. So you should be telling us some things about your response (DV).
The first point to make is that in ANOVA the marginal distribution of the response isn't assumed to be normal; it's the conditional distribution. How are you coming to the conclusion that your data are non-normal (how are you identifying the distribution?), and how non-normal are we talking?
The second point is that the importance of normality changes with sample size, yet you don't mention total sample size.
In using the bootstrap, you will need to regard some collection of quantities as exchangeable. In a two-way ANOVA this would normally be some form of residual but for residuals to be exchangeable, you need (for example) the variance of and the shape of the distribution not to change with the mean. These considerations would usually rule out applying it to count data, for example.
You do have alternatives; for some kinds of data you might consider applying a GLM to fit an ANOVA-like model -- I believe that is something you can do in SPSS.
Edit in response to the additional information in your edit to your question:
"Percentage of items correct" is a count (number of items correct) divided by a fixed total (number of items). Scaling aside, this is count data of a sort, for which ANOVA would not normally be appropriate, since you won't likely have linearity (because the response is bounded above and below, though this will only affect the size of the interaction in your case), and the equal variance assumption won't hold across different means (variance must vary as a function of the mean, because of the bounds).
an arcsin square root transformation would help to stabilize the variance, but it will not "make the data continuous" -- it's still just as discrete as before. It may help a little with the skewness, but it might not make much different.
There are models more suited to 2x2 count data (e.g. binomial GLMs, loglinear models, even chi-square tests) - but your data might not fit the usual models for counts because test questions are rarely of uniform difficulty and - even withing sub-groups - people are rarely of uniform ability. [You might try a binomial model and see if it's plausible. It's possible that a negative binomial model might be able to deal with the possible heterogeneity.]
If you're contemplating an arcsin square root transformation, it's usually an indication you should have used a GLM.
Best Answer
The assumption for a general linear model is that the data are marginally normal. That is, that the distributions of errors from the model are normally distributed. So, you want to take the residuals from the model, and assess those for normality.
I recommend against using a test to assess normality in the way you are suggesting. The problem is that these tests are sensitive to sample size and will find a significant deviation from normal for a large data set even if the deviation is small.
You are better off using visual methods. Your eyes and brain are a better judge. You can use a quantile-quantile plot or a histogram of residuals that you can compare to a normal distribution.