13.2 The f distribution and the f ratio

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This module describes how to calculate the F Ratio and F Distribution based on the hypothesis test for the One-Way ANOVA.

The distribution used for the hypothesis test is a new one. It is called the $F$ distribution, named after Sir Ronald Fisher, an English statistician. The $F$ statistic is a ratio (a fraction). There are two sets of degrees of freedom; one for the numerator and one forthe denominator.

For example, if $F$ follows an $F$ distribution and the degrees of freedom for the numerator are 4 and the degrees of freedom for the denominator are 10, then $F$ ~ ${F}_{4,10}$ .

The $F$ distribution is derived from the Student's-t distribution. One-Way ANOVA expands the $t$ -test for comparing more than two groups. The scope of that derivation is beyond the level of this course.

To calculate the $F$ ratio, two estimates of the variance are made.

1. Variance between samples: An estimate of ${\sigma }^{2}$ that is the variance of the sample means multiplied by n (when there is equal n). If the samples are different sizes, the variance between samples is weighted toaccount for the different sample sizes. The variance is also called variation due to treatment or explainedvariation.
2. Variance within samples: An estimate of ${\sigma }^{2}$ that is the average of the sample variances (also known as a pooled variance). When the sample sizes are different, thevariance within samples is weighted. The variance is also called the variation due to error or unexplained variation.
• ${\mathrm{SS}}_{\text{between}}=$ the sum of squares that represents the variation among the different samples.
• ${\mathrm{SS}}_{\text{within}}=$ the sum of squares that represents the variation within samples that is due to chance.

To find a "sum of squares" means to add together squared quantities which, in some cases, may be weighted. We used sum of squares to calculate the sample variance andthe sample standard deviation in Descriptive Statistics .

$\mathrm{MS}$ means "mean square." ${\mathrm{MS}}_{\text{between}}$ is the variance between groups and ${\mathrm{MS}}_{\text{within}}$ is the variance within groups.

Calculation of sum of squares and mean square

• $k$ = the number of different groups
• ${n}_{j}$ = the size of the $\text{jth}$ group
• ${s}_{j}$ = the sum of the values in the $\text{jth}$ group
• $n$ = total number of all the values combined. (total sample size: $\sum {n}_{j}$ )
• $x$ = one value: $\sum x=\sum {s}_{j}$
• Sum of squares of all values from every group combined: $\sum {x}^{2}$
• Between group variability: ${\text{SS}}_{\text{total}}=\sum {x}^{2}-\frac{{\left(\sum x\right)}^{2}}{n}$
• Total sum of squares: $\sum {x}^{2}-\frac{\left(\sum x{\right)}^{2}}{n}$
• Explained variation- sum of squares representing variation among the different samples ${\text{SS}}_{\text{between}}=\sum \left[\frac{\left(\text{sj}{\right)}^{2}}{{n}_{j}}\right]-\frac{\left(\sum {s}_{j}{\right)}^{2}}{n}$
• Unexplained variation- sum of squares representing variation within samples due to chance: ${\text{SS}}_{\text{within}}={\text{SS}}_{\text{total}}-{\text{SS}}_{\text{between}}$
• df's for different groups (df's for the numerator): ${\text{df}}_{\text{between}}=k-1$
• Equation for errors within samples (df's for the denominator): ${\text{df}}_{\text{within}}=n-k$
• Mean square (variance estimate) explained by the different groups: ${\text{MS}}_{\text{between}}=\frac{{\text{SS}}_{\text{between}}}{{\text{df}}_{\text{between}}}$
• Mean square (variance estimate) that is due to chance (unexplained): ${\text{MS}}_{\text{within}}=\frac{{\text{SS}}_{\text{within}}}{{\text{df}}_{\text{within}}}$

${\mathrm{MS}}_{\text{between}}$ and ${\mathrm{MS}}_{\text{within}}$ can be written as follows:

• ${\mathrm{MS}}_{\text{between}}=\frac{{\mathrm{SS}}_{\text{between}}}{{\mathrm{df}}_{\text{between}}}=\frac{{\mathrm{SS}}_{\text{between}}}{k-1}$
• ${\mathrm{MS}}_{\text{within}}=\frac{{\mathrm{SS}}_{\text{within}}}{{\mathrm{df}}_{\text{within}}}=\frac{{\mathrm{SS}}_{\text{within}}}{n-k}$

The One-Way ANOVA test depends on the fact that ${\mathrm{MS}}_{\text{between}}$ can be influenced by population differences among means of the several groups. Since ${\mathrm{MS}}_{\text{within}}$ compares values of each group to its own group mean, the fact that group means might be different doesnot affect ${\mathrm{MS}}_{\text{within}}$ .

The null hypothesis says that all groups are samples from populations having the same normal distribution. The alternate hypothesis says that at least two of the samplegroups come from populations with different normal distributions. If the null hypothesis is true, ${\mathrm{MS}}_{\text{between}}$ and ${\mathrm{MS}}_{\text{within}}$ should both estimate the same value.

The null hypothesis says that all the group population means are equal. The hypothesis of equal means implies that the populations have the same normal distribution because it is assumed that the populations are normal and that they have equal variances.

F-ratio or f statistic

$F=\frac{{\mathrm{MS}}_{\text{between}}}{{\mathrm{MS}}_{\text{within}}}$

If ${\mathrm{MS}}_{\text{between}}$ and ${\mathrm{MS}}_{\text{within}}$ estimate the same value (following the belief that ${H}_{o}$ is true), then the F-ratio should be approximately equal to 1. Mostly just sampling errorswould contribute to variations away from 1. As it turns out, ${\mathrm{MS}}_{\text{between}}$ consists of the population variance plus a variance produced from the differences between thesamples. ${\mathrm{MS}}_{\text{within}}$ is an estimate of the population variance. Since variances are always positive, if the null hypothesis is false, ${\mathrm{MS}}_{\text{between}}$ will generally be larger than ${\mathrm{MS}}_{\text{within}}$ . Then the F-ratio will be larger than 1.However, if the population effect size is small it is not unlikely that ${\mathrm{MS}}_{\text{within}}$ will be larger in a give sample.

The above calculations were done with groups of different sizes. If the groups are the same size, the calculations simplify somewhat and the F ratio can be written as:

F-ratio formula when the groups are the same size

$F=\frac{n\cdot {{s}_{\stackrel{_}{x}}}^{2}}{{{{s}^{2}}_{\text{pooled}}}^{}}$

Where ...

• $n=$ the sample size
• ${\text{df}}_{\text{numerator}}=k-1$
• ${\text{df}}_{\text{denominator}}=n-k$
• ${{s}^{2}}_{\mathrm{pooled}}=$ the mean of the sample variances (pooled variance)
• ${{s}_{\overline{x}}}^{2}=$ the variance of the sample means

The data is typically put into a table for easy viewing. One-Way ANOVA results are often displayed in this manner by computer software.

Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) F
Factor
(Between)
SS(Factor) k - 1 MS(Factor) = SS(Factor)/(k-1) F = MS(Factor)/MS(Error)
Error
(Within)
SS(Error) n - k MS(Error) = SS(Error)/(n-k)
Total SS(Total) n - 1

Three different diet plans are to be tested for mean weight loss. The entries in the table are the weight losses for the different plans. The One-Way ANOVA table is shown below.

Plan 1 Plan 2 Plan 3
5 3.5 8
4.5 7 4
4 3.5
3 4.5

One-Way ANOVA Table: The formulas for SS(Total), SS(Factor) = SS(Between) and SS(Error) = SS(Within) are shown above. This same information is provided by the TI calculator hypothesis test function ANOVA in STAT TESTS (syntax is ANOVA(L1, L2, L3) where L1, L2, L3 have the data from Plan 1, Plan 2, Plan 3 respectively).

Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) F
Factor
(Between)
SS(Factor)
= SS(Between)
=2.2458
k - 1
= 3 groups - 1
= 2
MS(Factor)
= SS(Factor)/(k-1)
= 2.2458/2
= 1.1229
F =
MS(Factor)/MS(Error)
= 1.1229/2.9792
= 0.3769
Error
(Within)
SS(Error)
= SS(Within)
= 20.8542
n - k
= 10 total data - 3 groups
= 7
MS(Error)
= SS(Error)/(n-k)
= 20.8542/7
= 2.9792
Total SS(Total)
= 2.9792 + 20.8542
=23.1
n - 1
= 10 total data - 1
= 9

The One-Way ANOVA hypothesis test is always right-tailed because larger F-values are way out in the right tail of the F-distribution curve and tend to make us reject ${H}_{o}$ .

Notation

The notation for the F distribution is $F$ ~ ${F}_{\text{df(num)},\text{df(denom)}}$

where $\text{df(num)}={\mathrm{df}}_{\text{between}}$ and $\text{df(denom)}={\mathrm{df}}_{\text{within}}$

The mean for the F distribution is $\mu =\frac{\mathrm{df\left(num\right)}}{\mathrm{df\left(denom\right)}-1}$

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