# Does SSTR (sum of squares for treatments) = SSR (regression sum of squares)?

I'm on my first course into statistics and there seems to be something in common for Regression and ANOVA analysis:

in ANOVA I'm told that SST (total sum of squares) = SSE(error sum of squares) + SSTR (sum of squares for treatments)

while in regression, SST (total sum of squares) = SSE(error sum of squares) + SSR (regression sum of squares)

Does that mean [after algebraic manipulation from the above 2 equations] SSR = SSTR ?

I've looked up the formula for both SSR and SSTR and they are as followed:

SSR=

SSTR=

Those two formulas look different. Does that mean I should not mix regression with ANOVA?

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As Jonathan says you are indeed correct - well spotted. Basically if you have a 1-factor ANOVA with say q levels, then you have observations indexed by $j$ and $k$: $Y_{jk}$, $j=1,....,q,k=1,...,n_{q}$. The ANOVA model is $Y_{jk}=\mu_{j}+\epsilon_{jk}$, where $\mu_{j}$ represents the unknown true factor levels you want to estimate. This model can be written in "regression" notation by indexing your variables with just one index, say $i$: $Y_{i}=\sum\nolimits_{j=1}^{q}x_{ij}\beta_{j}+\epsilon_{i}$, $i=1,....,N$, where $x_{ij}=1$ for one of the $j's$ and zero otherwise. We assume that for your $N$ regression observations we have $n_{j}$ observations with $x_{ij}=1$, and that $\sum\nolimits_{j=1}^{q}=N$. Letting $\hat{Y_{i}}=\sum\nolimits_{j=1}^{q}x_{ij}\hat{\beta_{j}}$ we see that

$SSR=\sum\nolimits_{i=1}^{N}(\hat{Y_{i}}-\bar{Y})^{2}=\sum\nolimits_{i=1}^{N}(\sum\nolimits_{j=1}^{q}x_{ij}\hat{\beta_{j}}-\bar{Y})^{2}=\sum\nolimits_{j=1}^{q}n_{j}(\hat{\beta}_{j}-\bar{Y})^{2}$.

Now due to $x_{ij}$ being zero or one we find that $\hat{\beta}_{j}=\bar{Y}_{j}$ (I mean this to denote the average of the $Y$'s where $x_{ij}=1$), thus $\sum\nolimits_{i=1}^{N}(\hat{Y_{i}}-\bar{Y})^{2}=\sum\nolimits_{j=1}^{q}n_{j}(\bar{Y}_{j}-\bar{Y})^{2}$. In ANOVA notation we have $\bar{Y}_{j}=\bar{Y}_{j\cdot}$, and so

$SSR=\sum\nolimits_{i=1}^{N}(\hat{Y_{i}}-\bar{Y})^{2}=\sum\nolimits_{j=1}^{q}n_{j}(\bar{Y}_{j\cdot}-\bar{Y})^{2}=SSTR$.

Basically ANOVA is just a restricted form of regression, the restriction being the variables are factor variables rather than continuous ones. I find it much easier to learn about regression first, and to then think of ANOVAs in this way. This is because all the theory of regression carries over to ANOVA, but the theory about the sums of squares of ANOVAs only applies to these specific regression models, and not to a general one (where continuous and factor variables are present). If you have the time it is worth learning about regression as well as ANOVAs since the theory of ANOVAs gets you thinking from a designed experiment viewpoint (randomised controlled trials), whilst regression theory is more general since it really is about you already having your data (not from a designed experiment) and wanting to analyze it.

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Is it that hard to spell my name right? :( – Jonathan Christensen Nov 15 '12 at 0:29
hold on a sec, your statement $SSR=\sum\nolimits_{i=1}^{N}(\hat{Y_{i}}-\bar{Y})^{2}=\sum\nolimits_{j=1}^{q}n_{‌​j}(\bar{Y}_{j\cdot}-\bar{Y})^{2}=SST$. means $SSR = SST$??!!! – user133466 Nov 16 '12 at 17:41
My apologies Jonathan for not spelling your name correctly - lazy typing on my part. Also to user133466, although I was not clear in my answer, I mean SST to be the sum of squares for the treatment effect in the ANOVA, rather than the total sum of squares - I should have used SSTR. I have now changed this. – dandar Nov 17 '12 at 17:27

SSR in regression is analogous to SSTR in ANOVA, yes. In particular, an ANOVA can be written as a special case of multiple regression, and your SSTR for that ANOVA is precisely the SSR of the corresponding regression.

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