Analysis of Variance - One Way

1. Introduction
2. Logic
3. Notation
4. Terminology
5. Partitioning the Variance
6. The F Test
7. Formal Example
   1. Research Question
   2. Hypotheses
   3. Assumptions
   4. Decision Rules
   5. Computation - [Minitab]
   6. Decision
8. Comparisons Among Means - [Minitab] [Spreadsheet]
9. Relation of F to t

Homework

I. Introduction

The ANalysis Of VAriance (or ANOVA) is a powerful and common statistical procedure in the social sciences. It can handle a variety of situations. We will talk about the case of one between groups factor here and two between groups factors in the next section.

The example that follows is based on a study by Darley and Latané (1969). The authors were interested in whether the presence of other people has an influence on whether a person will help someone in distress. In this classic study, the experimenter (a female graduate student) had the subject wait in a room with either 0, 2, or 4 confederates. The experimenter announces that the study will begin shortly and walks into an adjacent room. In a few moments the person(s) in the waiting room hear her fall and complain of ankle pain. The dependent measure is the number of seconds it takes the subject to help the experimenter.

How do we analyze this data? We could do a bunch of between groups t tests. However, this is not a good idea for three reasons.

1. The amount of computational labor increases rapidly with the number of groups in the study.
   
   

   Number Groups	Number Pairs of Means
   3	3
   4	6
   5	10
   6	15
   7	21
   8	28

2. We are interested in one thing -- is the number of people present related to helping behavior? -- thus it would be nice to be able to do one test that would answer this question.

3. The type I error rate rises with the number of tests we perform.

II. Logic

The reason this analysis is called ANOVA rather than multi-group means analysis (or something like that) is because it compares group means by analyzing comparisons of variance estimates. Consider:

We draw three samples. Why might these means differ? There are two reasons:

1. Group Membership (i.e., the treatment effect or IV).
2. Differences not due to group membership (i.e., chance or sampling error).

The ANOVA is based on the fact that two independent estimates of the population variance can be obtained from the sample data. A ratio is formed for the two estimates, where:

one is sensitive to ®		treatment effect & error		between groups estimate
and the other to ®		error		within groups estimate

Given the null hypothesis (in this case H_O: m₁=m₂=m₃), the two variance estimates should be equal. That is, since the null assumes no treatment effect, both variance estimates reflect error and their ratio will equal 1. To the extent that this ratio is larger than 1, it suggests a treatment effect (i.e., differences between the groups).

It turns out that the ratio of these two variance estimates is distributed as F when the null hypothesis is true.

Note:

1. F is a family of distributions, which varies as a function of a pair of degrees of freedom (one for each variance estimate).
2. F is positively skewed.
3. F ratios, like the variance estimates from which they are derived, cannot have a value less than zero.

Using the F, we can compute the probability of the obtained result occurring due to chance. If this probability is low (p £ a), we will reject the null hypothesis.

III. Notation

We already knew that:

i = any score
n = the last score

What is new here is that:

j = any group
p = the last group

Thus:

Group
1	2	J	P
X11	X12	X1j	X1p
X21	X22	X2j	X2p
Xi1	Xi2	Xij	Xip
Xn1	Xn2	Xnj	Xnp
T1	T2	Tj	Tp
n1	n2	nj	np

And:

1.
2.
3.
4.
5.

IV. Terminology

Since we are talking about the analysis of the variance, let's review what we know about it.

So the variance is the mean of the squared deviations about the mean (MS) or the sum of the squared deviations about the mean (SS) divided by the degrees of freedom.

V. Partitioning the Variance

As noted above, two independent estimates of the population variance can be obtained. Expressed in terms of the Sum of Squares:

To make this more concrete, consider a data set with 3 groups and 4 subjects in each. Thus, the possible deviations for the score X₁₃ are as follows:

As you can see, there are three deviations and:

total	within groups	between groups
#3	#1	#2

To obtain the Sum of the Squared Deviations about the Mean (the SS), we can square these deviations and sum them over all the scores.

Thus we have:

Note: n_j in formula for the SS_Between means do it once for each deviation.

VI. The F Test

It is simply the ratio of the two variance estimates:

As usual, the critical values are given by a table. Going into the table, one needs to know the degrees of freedom for both the between and within groups variance estimates, as well as the alpha level.

Example: If we have 3 groups and 10 subjects in each, then

DfB	= p - 1	= 3 – 1 = 2
DfW	= p(n - 1)      or = N - p	= 3 * (10-1) = 27      or = 30 - 3 = 27
DfT	= N - 1	= 30 - 1 = 29


Note that the df add up to the total and with a =.05, F_crit= 3.35

VII. Formal Example

1. Research Question
   Does the presence of others influence helping behavior?

2. Hypotheses
   
   

   	In Symbols	In Words
   HO	m1=m2=m3	The presence of others does not influence helping.
   HA	Not Ho	The presence of others does influence helping.

3. Assumptions
   1) The subjects are sampled randomly.
   2) The groups are independent.
   3) The population variances are homogenous.
   4) The population distribution of the DV is normal in shape.
   5) The null hypothesis.

4. Decision rules
   Given 3 groups with 4, 5, and 5 subjects, respectively, we have (3-1=) 2 df for the between groups variance estimate and (3+4+4=) 11 df for the within groups variance estimate. (Note that it is good to check that the df add up to the total.) Now with an a level of .05, the table shows a critical value of F is 3.98. If F_obs ³ F_crit, reject H_o, otherwise do not reject H_o.
5. Computation - [Minitab]
   

   Here is the data (i.e., the number of seconds it took for folks to help):

   
   

   	# people present
   	0	2	4
   	25	30	32
   	30	33	39
   	20	29	35
   	32	40	41
   		36	44
   	107	168	191
   	4	5	5
   	26.8	33.6	38.2

   A good way to describe this data would be to plot the means:
   
   

For the analysis, we will use a grid as usual for most of the calculations:

	0	X2	2	X2	4	X2
	25	625	30	900	32	1024
	30	900	33	1089	39	1521
	20	400	29	841	35	1225
	32	1024	40	1600	41	1681
			36	1296	44	1936
	107		168		191		466	T
	4		5		5		14	N
	26.8		33.6		38.2
		2949		5726		7387	16062	II
	2862.25		5644.8		7296.2		15803.25	III

Now we need the grand totals and the three intermediate quantities:

I.
II.
III.

And:

SS_B = III – I = 15803.25-15511.14 = 292.11
SS_W = II – III = 16062-15803.25 = 258.75
SS_T = II – I = 16062-15511.14 = 550.86

Thus:

Source	SS	df	MS	F	p
Between	292.11	2	146.056	6.21	<.05
Within	258.75	11	23.520
Total	550.86	13

6. Decision
   Since F_obs (6.21) is > F_crit (3.98), reject H_o and conclude that the more people present, the longer it takes to get help.

VIII. Comparisons Among Means

In the formal example presented above, we rejected the null and asserted that the groups were drawn from different populations. But which groups are different from which? This leads to a distinction between two types of comparisons that we can perform: preplanned versus post hoc.

Preplanned	Post hoc
We have a theory (or some previous research) which suggests certain comparisons.	Have a significant overall (or omnibus) F & then want to localize the effect.
In this case, we might not even compute the omnibus F (this approach is somewhat analogous to the one-tailed test).	Are more commonly used than preplanned comparisons.

With three groups (Groups 1, 2 & 3), the following 6 comparisons are possible.

Simple	Complex
1 vs. 2 1 vs. 3 2 vs. 3	(1 + 2) vs. 3 1 vs. (2 + 3) (1 + 3) vs. 2

This leads to another categorization of comparisons into two types:

1. Simple - between pairs of means (illustrated by the first 3 comparisons above).
2. Complex - involving more than two means (illustrated by the last 3 comparisons above).

As the number of groups increases, so does the number of comparisons that are possible. Some of these can tell us about trend (a description of the form of the relationship between the IV and DV).

The problem with post hoc tests is that the type I error rate increases the more comparisons we perform. This is a somewhat controversial area and there are a number of methods currently in use to deal with this problem. We will consider one of the more simple methods below.

The protected t test - [Minitab] [Spreadsheet]

• It is performed only when the omnibus F is significant.
• It uses a more stable estimate of the population variance than the t test (i.e., instead of and as a result the df is greater.
• This technique is protected because it requires the omnibus F to be significant (which tells us there is at least one comparison between means that is significant). So, in other words, it is protected because we are not just shooting in the dark.

The formula is:

(Where the dfs are 1 for the numerator and df_w for the denominator.)

So, for our example the critical value of F is 4.84 (from the table) and:

Thus, the only comparison that is significant is that between the first and third groups.

IX. Relation of F to t

Since the F test is just an extension of the t test to more than two groups, they should be related and they are.

With two groups, F = t² (and this applies to both the critical and observed values).

For example, consider the critical values for df = (1, 15) with a = .05:

F_{crit (1, 15)} = t_{crit (15)}²

Obtaining the values from the tables, we can see that this is true, that is:

4.54 = 2.131²