# What are the problems with doing regression on sample means instead of ANOVA in a time series setting?

I have data from an annual survey which seeks to assess the needs of consumers on a 1 (least important) to 5 (most important) scale. Thus, I have data that looks like so:

My goal is to assess if the needs of consumers have changed across years and thus I believe the 'correct' approach to assess if needs have changed is to do a one-way ANOVA with year being the factor.

However, another approach is to consider the sample means across years like shown in the table below:

We can then consider a linear regression of RAM vs Years and check if the slope is significantly different from 0. If it is not significantly different from 0 then we can perhaps conclude that the observed variation in customer needs across years is due to random fluctuation and not due to any underlying shifts.

What are the drawbacks of the second approach relative to the first? Are they perhaps addressing different questions? Is the second one a weaker test of changes in customer needs as we lose sample information?

ANOVA is regression and regression is ANOVA. Here are both models in terms of matrix algebra:

$Y = XB + e$

where Y is a vector of your DV values, X is a matrix of IV values, B is a vector of parameters to be estimated and e is a vector of error terms.

Both assume that $e \sim \mathcal{N}(\mu, \sigma)$ and that the errors are independent and identically distributed. As @Abs first answer points out, these assumptions are violated, so the general linear model (the term for both ANOVA and regression) is inappropriate. You should use a model that accounts for this, such as a multilevel model.

• I agree with your response on the assumptions but ignoring those issues for the moment what is your assessment of the issues with the two different methods of analysis. – stats_student Oct 22 '12 at 1:54
• They aren't two methods, they are one method. – Peter Flom - Reinstate Monica Oct 22 '12 at 10:15
• Yes, I agree they are but the two approaches I outlined in my question are: using ANOVA on raw data and regression on sample means. Since the data being used is different, the results will be different and hence the question. If I were using ANOVA and regression on the same data then your point is well taken. But, that is not the focus of my question. – stats_student Oct 22 '12 at 14:36

I believe that non of them are suitable in your case. In both of them, samples have to be independent. Moreover, ANOVA procedure is robust if dependent variable is approximately normal. So, I think that it's better to use non parametric methods such as Kruskal–Wallis one-way analysis of variance.

• Thanks for your answer but ignoring the issues (lack of normality, dependent samples etc) you mentioned what would be the differences/trade-offs between the two methods? – stats_student Oct 19 '12 at 14:25
• If you do regression on just the three points in the bottom table you have junk. That's a way overfitted model that doesn't account for the variation in the data. That's not allowed. – Peter Flom - Reinstate Monica Oct 22 '12 at 15:37

If we ignore the issues, it sounds that ANOVA would work better. I think linear regression is not suitable for your case (you have only a few values for independent variable). Moreover, linear regression can not figure out all differences. For instance, it's possible to have a non significant slop in linear regression but, there exist significant difference between only two years. So, it's better to consider year variable as a factor (not an explanatory variable) and carry out ANOVA.