# Link between the null hypothesis and test statistics in F-test

How are $H_0$ and $H_1$ linked to the F value in the following cases? I understand how each of $H_0$ and $H_1$ gets more likely depending on F-value, yet I was trying to find more of a mathematical link between the hypotheses and the F-value (Not the derivation of F-distribution). I am trying to understand the following usages of F-test from the perspective of F-test for equality of variances; I want to confirm $H_0^F, H_1^F$ are equivalent to $H_0, H_1$ respectively.

Following are two uses of F-tests other than the equality of variance test.
Use1: Equality of means with equal variance assumption (ANOVA):
sample statistics: $$F_{obs} = \dfrac{MS_B}{MS_W} = \dfrac{\sum_{i=1}^N{(\bar{y}_{g_i} - \bar{y})^2}/(K-1)}{\sum_{i=1}^N{(\bar{y}_{g_i} - y_i)^2}/(N-K)}$$ $H_0: \mu_1 = \mu_2 = \mu_3$
$H_1: \mu_i \neq \mu_j$ for a pair $i \neq j, (i,j) \in \{1,2,3\}$

[Question for Use1]
Is my $H_0^F,H_1^F$ below correct?
If so, are $H_0, H_1$ above being mathematically derived from and equivalent to $H_0^F,H_1^F$ respectively: $$H_0^F: \sigma^2_B = \sigma^2_W, \ H_1^F: \sigma^2_B \neq \sigma^2_W$$

where $MS_B$ is the sample estimate for $\sigma^2_B$ and $MS_W$ the sample estimate for $\sigma^2_W$.

Use2: Comparison of fits in two nested OLS models
sample statistics:
$$F_{obs} = \dfrac{\dfrac{SSE(R) - SSE(F)}{df_R - df_F}}{\dfrac{SSE(F)}{df_F}}$$

$H_0:$ Fit of the reduced model and the full model is equal.
$H_1:$ Fit of the full model is better than that of the reduced model.

[Question for Use2]
What is $H_0, H_1$ in math not in english?
Is my $H_0^F,H_1^F$ below correct?
If so, are $H_0, H_1$ above being derived from and equivalent to $H_0^F,H_1^F$ respectively: $$H_0^F: \sigma^2_{Reduced} - \sigma^2_{Full} = \sigma^2_{Full}, \ H_1^F: \sigma^2_{Reduced} - \sigma^2_{Full} \geq \sigma^2_{Full}$$

where $\dfrac{SSE(R) - SSE(F)}{df_R - df_F}$ is the sample estimate for $(\sigma^2_{Reduced} - \sigma^2_{Full})$ and $\dfrac{SSE(F)}{df_F}$ the sample estimate for $\sigma^2_{Full}$.

There is a difference between the observed/estimated means $\hat{\mu}_1$, $\hat{\mu}_2$ and $\hat{\mu}_3$ and the population/true means $\mu_1=\mu_2=\mu_3$.

The observed means are not expected to be the same. Because of randomness in sampling you will expect to see variations.

In fact if the observed means (and the same is true for the comparisons of observed models) are too much the same then this may indicate something is wrong with the sampling (possibly there is some bias, e.g. a scientists that is selective, see for example Fisher about Mendel's experiments and data).

$$\begin{array}\\H_0:&\mu_1 = \mu_2 = \mu_3 \qquad or \\ H_0:& \text{ the reduced model equals the full model}\\&\text{(ie the additional model terms are zero)}\end{array}$$

These only relate to the population. In the sample you will expect to observe differences which can be expressed by separating the variance of the observed variable into components which is what analysis of variance is about and can be quantified by the F statistic:

$$Var(Y_{total}) = Var(Y_1) + Var(Y_2) + ... + Var(Y_n)$$

Examples are like yours:

1. random variations between groups + random variations within groups
2. random variations of model1 + random variations of model2-model1 + random variations in an error term

Then the F-test relates to the relative contributions of those variances, scaled by the degrees of freedom. If the groupings or models have no meaning (are just random intersections of the data) then in general those contributions to the variance should be equal (not large not small). If the starting point is your null hypothesis then we look at a one-sided F-test (when we are only testing whether a certain variance is exceeding a limit, meaning a falsification of the $H_0$ and we are not interested in an extremely small variation).

This analysis of variance has a wider applicability and can be more nuanced.

The nuance is strongest in your case 1. The $\hat{\mu}_i$ can be considered means of the sample groups. However we are not always so much interrested in the actual difference between the $\hat{\mu}_i$.

Actually the population $\mu_i$ might be considered different coming from a larger population distribution with the same $\mu$ (for instance in a random effects model). In this case one does not necessarily wish to analyze $\mu_1=\mu_2$ and possibly more something like whether the variation between the different $\mu_i$ is a larger or smaller component in the variation of some parameter.

Example: If we collect data from multiple students from a couple of schools. Then surely the results grouped by school may likely be resulting in different means (there may not even be a difference between estimate and real value, $\mu_i = \hat{\mu}_i$, if we sample all the students from a school), and not what the $H_0$ is about. Such differences may be considered as a (expected) random effect (the schools being a sample from a larger population of schools) and not relevant for the study.

So I'd say that the F-test, comparing sizes of variations in different variables with different degrees of freedom, is more general than the null hypothesis that you wrote down. But in your cases F-test based on the F-statistic and it's expected distribution can be derived from the $H_0$.

• In case 1 if the $\mu_i$ are equal in the population, then the $\hat{\mu}_i$ in the sample can be different, but follow a distribution around the same $\mu$ with a variance in accordance with the variations within the groups. (I hope you saw the nuance, that we do not hypothesize that sample means are gonna be the same, but instead that the population means are gonna be the same)
• In case 2 you get something similar. The measured difference in variance between the full and reduced model is not expected to be zero. We only expect this to be the case for the population. When we take a sample then we will expect some random correlation of the model components causing some change in variance between the full and reduced model.
• Hmm, I already understand that the hypotheses of F-test concern the population statistics and that the sample statistics may differ for groups. It is my mistake that I misused the sample statistics such as mean square (MS) in the hypotheses in OP. I have edited it. – villybyun Dec 13 '17 at 2:12
• @villybyun, now I see that this difference is not the main problem I will streamline my answer a bit more to be less about that. But still, this consideration is the keypoint in the derivation of going from your initial hypothesis to the associated hypotheses in the F-test about the F-statistic. Namely, under your null hypotheses you will expect the F-statistic to be below a maximal value. – Sextus Empiricus Dec 13 '17 at 9:05
• (The low end of the F-distribution is not interesting if you test against the alternative hypothesis that the groups or models are different, in which case the low end of the F-distribution is even less likely). The one-sided F-test is a more powerful test. en.wikipedia.org/wiki/Neyman%E2%80%93Pearson_lemma – Sextus Empiricus Dec 13 '17 at 9:07
• Regarding your comment "derivation of going from your initial hypothesis to the associated hypotheses in the F-test about the F-statistic", do you have any reference (online, book) that I can look up that process? In other words, are my $H_0^F, H_1^F$ equal to $H_0, H_1$ in the two cases and how? – villybyun Dec 13 '17 at 19:01