# Two-way ANOVA for data with homogeneity of variance is violated

I'm currently doing a biomechanical tissue study and want to run a two-way ANOVA.

Here is how my data look like:

• Independent variable #1 : Treatment or Non-treatment
• Independent variable #2 : Different 3 conditions
• Dependent variable : Elasticity of tissues (n=23)

At first, my data showed not a normality, so I transformed to log10 and became good normal distribution. However, Levene's test showed that the data violated homogeneity of variance. The sample sizes in three different conditions are all the same. But in condition 3, the standard deviation is pretty large; assuming this makes data heterogeneous. To handle this, I also checked the ratio of largest to smallest variations, showing 5.45 which is higher than 3~4 which is a rule of thumb. In this case, what can I do more to run a two-way ANOVA? or if I have to run alternative test (non-parametric), what options do I have?

ps. Following is the result from SPSS.

Treatment Group

 Condition  Average       SD       N
1       -1.7416    0.11994     23
2       -1.7766    0.10215     23
3       -1.4389    0.21877     23
Total:    -1.6524    0.21642     69


Non-treatment Group

 Condition  Average       SD       N
1       -1.6356    0.09080     23
2       -1.6270    0.16576     23
3       -1.3469    0.16575     23
Total:    -1.5362    0.19705     69


Levene's test : F 6.488 / df1 5 / df2 132 / p<0.000

## 1 Answer

At first, my data showed not a normality, so I transformed to log10 and became good normal distribution.

Note that this automatically transforms the effects. See this question on the same transformation you did. You would have to discuss if you rather want a generalized linear model (not "general linear model").

It is generally not a good idea to transform in order to keep the distributional assumptions. If the assumptions are not met, you will miss your type-I-error rate, which is bad, but is healed often by large sample sizes. On the other hand, if you transform it in a way that you cannot interpret the effects in terms of your experiment, it is worse.

Variance homogeneity (homoskedasticity) is usually not so big a problem any more as there are procedures which can deal with it (I think they are even implemented in SPSS). In your case, data are even perfectly balanced (N=23), so there is even an exact procedure for it. I would generally suggest to use a procedure supporting homoskedasticity. If your choice of method depends on the result of a variance homogenity test, you are in fact using a two-staged procedure that behaves differently than the single staged procedures.

• @Jeremy The variance differences here are reasonably moderate, and variance heterogeneity is also somewhat less important with a balanced design (as here), though if you're doing post hoc contrasts or pairwise multiple comparisons it depends on which precise effects are of interest. There's some indication of changing spread with level after taking logs which suggests there may be a variance function that increases slightly faster even than mean-squared. – Glen_b Sep 27 '16 at 5:26