Inferential statistics - F ratios and the F-test

Ed 510 Applications of Educational Research

Tests for statistical differences in more than two sample means.
Calculated means can be observably different. But are these differences likely to occur by chance alone?
In other words - is it true that three means are statistically equal? Or four or five means, depending on the number of groups that are being compared?
One-way analysis of variance will evaluate the statistical differences between sample means when there is one independent variable.

Example - A test of social responsibility

A social scientist has constructed three groups of students. Each group consists of 3 experimental subjects and 1 to 3 additional members who are really stooges of the experimenter. A fire alarm is called. The dependent variable is the time that elapses before someone in any of the 3 groups decides to call the university security office. The independent variable is the stooge condition (varied number of stooges).

The scores for each group are as follows

Group 1 Group 2 Group 3
1 stooge 2 stooges 3 stooges

8                                9                            16
4                               10                            18
3                               14                            20

Calculate the sample mean for each group. Determine whether they are different. Then calculate the mean for the entire set of scores.

Group 1 mean _______________

Group 2 mean________________

Group 3 mean_______________

The mean of means ___________(The grand mean)

What makes this different in focus from a t-test is the following: Although two independent sample means can be compared to each other simultaneously, three sample means cannot. Therefore the mean of means (for the entire set of scores) is the reference point for comparison. Each sample mean, when compared to the mean of means, will be equal to, greater than, or less than the mean of means. When any two sample means are different from the mean of means, the question then is whether they are different from each other.

How are these differences tested statistically?

Instead of comparing means, variances are compared.

Types of variance can be summarized as follows.

Variability between groups that are dependent on group (treatment, intervention, etc.) BETWEEN GROUPS
Variability within groups (how individuals vary within each cell condition) WITHIN GROUPS
Variability over the entire design (the variability of groups combined with the variability of individuals) TOTAL

There are three different statistics that must be computed for each source of variation. Each statistic is computed three times: for between groups; for within groups and for the total, respectively.

These statistics are

The SS (sum of squares) - which represents the total variability in between, within and total sources of variation.
The df (degrees of freedom) - which are estimates of population size for the between, within and total sources of variation
The MS (mean squares) - which represents the variance for between, within and total sources of variation

Mean squares are always the result of dividing the SS by the df.

The table below helps to visualize how these statistics fit together. It is referred to as a source table. Once all the values for SS, df and MS are calculated then they are inserted into the appropriate cells in the source table.

Source of variation	Sum of Squares SS	degrees of freedom df	Mean Squares MS
Between groups
Within groups
Total

When the Mean Squares (MS) are calculated, Sum of Squares is always divided by df.

A. Between groups or SS between groups (i.e., total variation between groups)/ df between = MS between groups

B. Within groups or SS within groups (i.e, total variation within groups)/df within =MS within groups.

_{and
Total variation or SS total (i.e., total variation between and within
groups/df total = MS total.}

Each type of df is calculated separately: e.g., Groups - 1 _between; N-Groups-1 _within; and N-1 _total.

Practice computing MS's for the following sets of numbers

SS between = 100 and df = 2 S within = 500 and df = 50 SS total = 500 and df = 52

then MS would be then MS would be
then MS would be

The F ratio is computed by dividing the MS _betweenby the MS _within

For example: if MS _between is 500 and MS _within; then the value of the F-ratio is ? (That' right, 50)

Try these two short exercises. Fill in the blanks below.

There are four groups, each group represents a class that has learned a different study strategy to improve reading comprehension. The independent variable is _________________. The dependent variable would probably be __________. There are 100 subjects, 25 in each group.

What then are the steps in calculating an F-test?

Calculate the SS between groups
Calculate the df between groups
Calculate the MS between groups

then

Calculate the SS within groups
Calculate the df within groups
Calculate the MS within groups

then

Calculate the SS total
Calculate the df total
Calculate the MS total

then

Divid the MS between groups by the MS within groups, giving you the F-ratio

Now complete the source table

Source of variation	Sum of squares	df	Mean squares	F -ratio
Between groups	1000.00
Within subjects	500.00
Total

The calculated F-ratio is then compared to a tabled value, or critical value, of F to determine whether the groups are in deed statistically different and whether the alternative hypothesis should be supported.

In class we will practice reading computer printouts and using tabled values of F to further explore the F test.

Summarizing questions

Can you think of three different experiments you might design that would employ the F-test.

Page created January 5, 2001. Page modified January 20, 2001. Copyright Antonia D'Onofrio 2001/2002/2003.

SS between = 100 and df = 2	S within = 500 and df = 50	SS total = 500 and df = 52
then MS would be	then MS would be	then MS would be