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

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.