1 Psych 5500/6500 Introduction to the F Statistic (Segue to ANOVA) Fall, 2008

2 Overview of the F test The F test is used in many contexts. We will begin by taking a general look at how the F test works. In its most general form, the F test is used to determine whether two populations have the same variance.

3 Example We know that the mean height of men is greater than the mean height of females, but what about their respective variances? (two-tail example) H0: σ² Female = σ² Male HA: σ² Female  σ² Male

4 Test Statistic The test statistic is: If H0 is true, then both estimates are independently estimating the same thing, and thus should roughly equal each other (they won’t exactly equal each other due to random bias), and thus if H0 is true then the value of F obt should be around 1.

5 Degrees of Freedom There are two different degrees of freedom in the F test, one for the numerator and one for the denominator. Remember that: The numerator has df 1 = N 1 -1 and the denominator has df 2 = N 2 -1

6 Expected (Mean) Value of F Again, if H0 is true and both populations have the same variance then we would expect est.σ² 1 to approximately equal est.σ² 2 and thus F should be around 1. Random bias in the value of the denominator has a strange effect on the value of F, however, when est.σ² 2 is way larger than est.σ² 1 it drives the value of F from 1 down towards 0, but when est.σ² 2 is way smaller than est.σ² 1 it drives the value of F from 1 up towards infinity. The end result of this is that...

7 Expected (Mean) Value of F Remember that df 2 is the df for est.σ² 2 (i.e. N 2 -1). Thus if sample 2 has 30 scores in it, then df 2 would equal 29, and the mean value of F when H0 is true would be:

8 Expected (Mean) Value of F As the N of group 2 gets larger then est.σ² 2 becomes more accurate and the expected value of F gets closer to 1. For example, if N 2 = 500 then when H0 is true μ F = Bottom line: if H0 is true then est.σ² 1  est.σ² 2 and μ F  1 rather than the more intuitively reasonable μ F =1.

9 Sampling Distribution Now that we have a test statistic (F) we can look at the ‘Sampling Distribution of F assuming H0 is true’. The mean value of F will be close to 1...actually df 2 /(df 2 -2)...if H0 is true. The sample distribution is not a normal distribution, or a t distribution, it is not even symmetrical, as it has a mean close to ‘1’, but the lowest value F can take on is zero and the highest value is infinity.

10 Shape of the F distribution The shape of the F distribution is dependent upon the degrees of freedom of both the numerator and denominator. Red has df 1 =2 and df 2 =3, blue has df 1 = 4 and df 2 =30, and black has df 1 = 20 and df 2 =20.

11 Hypotheses Two-tail test: H0: σ² 1 = σ² 2 HA: σ² 1  σ² 2 One-tail test predicting σ² 1 < σ² 2 H0: σ² 1  σ² 2 HA: σ² 1 < σ² 2 One-tail test predicting σ² 1 > σ² 2 H0: σ² 1  σ² 2 HA: σ² 1 > σ² 2

12 F c values As the shape of the F distribution changes with different degrees of freedom, you need to know both df to find the Fc values. Remember: df 1 (i.e. for the numerator of F)= N-1 for est. σ² 1 df 2 (i.e. for the denominator of F) = N-1 for est. σ² 2

13 F c values Because of the way the F test is used in ANOVA (which we will get to later) F c tables rarely have the left-tail F c value. The F distribution tool I provide makes it easy to find the Fc values (enter a p of.975 and then a p of.025). The left-tail F c value can also computed fairly easily from a table that only has right-tail F c values.

14 Calculating Fc Left Tail Note the switch of df in the F c right tail. Example:

15 Back to Our Example We know that the mean height of men is greater than the mean height of females, but what about their respective variances? H0: σ² Female = σ² Male HA: σ² Female  σ² Male N Female =16, N Male =11 Set up the sampling distribution of F assuming H0 is true. μ F =10/8=1.25 if H0 true. Fc = 0.33 and 3.52

16 Sampling Distribution of F

17 Computations (by hand) Females: N Female =16, SS Female =46 Males: N Male =11, SS Male =25

18 Computations SPSS While SPSS doesn’t provide this use of the F test it will provide the ‘Variance’ of each group. Remember that in SPSS the ‘Variance’ of a group is actually the est. σ² of the population from which the sample was drawn, which is just what we need to compute F. You will still, however, need an F table to come up with the F critical values.

19 Decision H0: σ² Female = σ² Male HA: σ² Female  σ² Male If H0 is true then we would expect F to approximately equal If H0 is false we would expect F to not equal 1.25 In this case F obtained = 1.22, does this differ enough from what H0 predicted to reject H0? Mark the approximate location of F=1.22 on the ‘sampling distribution of F assuming H0 is true’ to see if you can reject H0. In this case we ‘do not reject H0’, we were unable to determine whether or not the two population variances differ.

20 One-Tail Test If we were testing a theory which predicted that women have a greater variance: H0: σ² Female  σ² Male HA: σ² Female > σ² Male We need to look up the one-tail F critical value (upper tail in this case). If H0 is true then we would expect F to be less than or equal to If H0 is false we would expect F to be greater than 1.25 (which is where we will put the rejection region).

21 Sampling Distribution of F

22 One-Tail Test If we were testing a theory which predicted that women have lesser variance: H0: σ² Female  σ² Male HA: σ² Female < σ² Male We need to look up the one-tail F critical value (lower tail in this case). If H0 is true then we would expect F to be greater than or equal to If H0 is false we would expect F to be less than 1.25 (which is where we will put the rejection region).

23 Sampling Distribution of F

24 Assumptions of this Use of F 1.The two variance estimates are independent of each other. 2.Both populations are normally distributed. Monte Carlo studies have shown that this assumption is quite important for the validity of this test.

25 Back to the Assumptions of the t Test One of the assumptions of the t test for independent means is that the variances of the two populations are equal. The F test we have just covered can test that assumption. But remember, due to the nature of null hypothesis testing, we can prove two variances are different but we can’t prove two variances are equal, because we can’t prove that H0 is true (unless we can show we have a powerful experiment, which would make beta small). The affect that non-normality has on the validity of this F test has led to it not being used as much as Levene’s test (mentioned next).

26 Levene’s Test Levene’s test is another way to determine whether or not the population variances are the same. Levene’s test has two advantages over the F test we just covered. First, it is less dependent upon the populations being normally distributed. Second, it can be used to test whether several groups all have the same variance. H0: σ² 1 = σ² 2 = σ² 3 … HA: at least one σ² is different than the rest. We will cover Levene’s test and how it works soon.