What if. . . You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days.

What if. . . You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days a person misses class) You would simply do a two-sample t-test two-tailed Easy!

But, what if. . . You were asked to determine if psychology, sociology, and biology majors have significantly different class attendance You can’t do a two-sample t-test You have three samples No such thing as a three sample t-test!

One-Way ANOVA ANOVA = Analysis of Variance
This is a technique used to analyze the results of an experiment when you have more than two groups

Example You measure the number of days 7 psychology majors, 7 sociology majors, and 7 biology majors are absent from class You wonder if the average number of days each of these three groups was absent is significantly different from one another

Hypothesis Alternative hypothesis (H1)
H1: The three population means are not all equal

Hypothesis Alternative hypothesis (H1) socio = bio

Hypothesis Alternative hypothesis (H1) socio = psych

Hypothesis Alternative hypothesis (H1) psych = bio

Hypothesis Alternative hypothesis (H1) psych = bio =  soc

Hypothesis Alternative hypothesis (H1)
Notice: It does not say where this difference is at!!

Hypothesis Null hypothesis (H0) psych = socio = bio
In other words, all three means are equal to one another (i.e., no difference between the means)

Results X = 3.00 X = 2.00 X = 1.00

Logic Is the same as t-tests
1) calculate a variance ratio (called an F; like t-observed) 2) Find a critical value 3) See if the the F value falls in the critical area

Between and Within Group Variability
Two types of variability Between / Treatment the differences between the mean scores of the three groups The more different these means are, the more variability!

Between Variability Compute S2 on the means X = 3.00 X = 2.00 X = 1.00

Between Variability S2 = 1 X = 3.00 X = 2.00 X = 1.00

Between Variability + 5 X = 3.00 X = 2.00 X = 1.00

Between Variability X = 8.00 X = 2.00 X = 1.00

Between Variability Compute S2 on the means X = 8.00 X = 2.00 X = 1.00

Between Variability S2 = 14.33 X = 8.00 X = 2.00 X = 1.00

Between Group Variability
What causes this variability to increase? 1) Effect of the variable (college major) 2) Sampling error

Two types of variability Within / Error the variability of the scores within each group

Within Variability Compute S2 within each group X = 3.00 X = 2.00

Within Variability S2 =.67 S2 =1.67 S2 =.67 X = 3.00 X = 2.00 X = 1.00

Within Group Variability
What causes this variability to increase? 1) Sampling error

Between-group variability Within-group variability

sampling error + effect of variable sampling error

sampling error + effect of variable sampling error Thus, if null hypothesis was true this would result in a value of 1.00

sampling error + effect of variable sampling error Thus, if null hypothesis was not true this value would be greater than 1.00

Calculating this Variance Ratio

Degrees of Freedom dfbetween dfwithin dftotal
dftotal = dfbetween + dfwithin

Degrees of Freedom dfbetween = k - 1 (k = number of groups)
dfwithin = N - k (N = total number of observations) dftotal = N - 1 dftotal = dfbetween + dfwithin

Degrees of Freedom dfbetween = k - 1 3 - 1 = 2
dfwithin = N - k = 18 dftotal = N = 20 20 =

Sum of Squares SSBetween SSWithin SStotal
SStotal = SSBetween + SSWithin

Sum of Squares SStotal

Sum of Squares SSWithin

Sum of Squares SSBetween

Sum of Squares Ingredients: X X2 Tj2 N n

To Calculate the SS

X Xs = 21 Xp = 14 XB = 7

X X = 42 Xs = 21 Xp = 14 XB = 7

X2 X = 42 Xs = 21 Xp = 14 XB = 7 X2s = 67 X2P = 38 X2B = 11

X2 X = 42 X2 = 116 Xs = 21 Xp = 14 XB = 7 X2s = 67 X2P = 38

T2 = (X)2 for each group X = 42 X2 = 116 Xs = 21 Xp = 14 XB = 7
T2P = 196 T2B = 49 T2s = 441

Tj2 X = 42 X2 = 116 Tj2 = 686 Xs = 21 Xp = 14 XB = 7 X2s = 67
T2P = 196 T2B = 49 T2s = 441

N X = 42 X2 = 116 Tj2 = 686 N = 21 Xs = 21 Xp = 14 XB = 7
T2P = 196 T2B = 49 T2s = 441

n X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7 Xs = 21 Xp = 14 XB = 7
T2P = 196 T2B = 49 T2s = 441

Ingredients X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7

Calculate SS X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7 SStotal

Calculate SS 42 32 116 21 SStotal X = 42 X2 = 116 Tj2 = 686 N = 21

Calculate SS X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7 SSWithin

Calculate SS 686 18 116 7 SSWithin X = 42 X2 = 116 Tj2 = 686 N = 21

Calculate SS X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7 SSBetween

Calculate SS 14 686 42 7 21 SSBetween X = 42 X2 = 116 Tj2 = 686

Sum of Squares SSBetween SSWithin SStotal
SStotal = SSBetween + SSWithin

Sum of Squares SSBetween = 14 SSWithin = 18 SStotal = 32 32 =

Calculating the F value

14 7 2

7

7 18 1 18

7 7 1

How to write it out

Significance Is an F value of 7.0 significant at the .05 level?
To find out you need to know both df

Degrees of Freedom Dfbetween = k - 1 (k = number of groups)
dfwithin = N - k (N = total number of observations)

Degrees of Freedom Dfbetween = k - 1 3 - 1 = 2
dfwithin = N - k = 18 Use F table Dfbetween are in the numerator Dfwithin are in the denominator Write this in the table

Critical F Value F(2,18) = 3.55 The nice thing about the F distribution is that everything is a one- tailed test

Decision Thus, if F value > than F critical
Reject H0, and accept H1 If F value < or = to F critical Fail to reject H0

Current Example F value = 7.00 F critical = 3.55
Thus, reject H0, and accept H1

Alternative hypothesis (H1)
H1: The three population means are not all equal In other words, psychology, sociology, and biology majors do not have equal IQs Notice: It does not say where this difference is at!!

How to write it out

Six Easy Steps for an ANOVA
1) State the hypothesis 2) Find the F-critical value 3) Calculate the F-value 4) Decision 5) Create the summary table 6) Put answer into words

Example Want to examine the effects of feedback on self-esteem. Three different conditions -- each have five subjects 1) Positive feedback 2) Negative feedback 3) Control Afterward all complete a measure of self-esteem that can range from 0 to 10.

Example: Question: Is the type of feedback a person receives significantly (.05) related their self-esteem?

Results

Step 1: State the Hypothesis
H1: The three population means are not all equal H0: pos = neg = cont

Step 2: Find F-Critical Step 2.1
Need to first find dfbetween and dfwithin Dfbetween = k (k = number of groups) dfwithin = N - k (N = total number of observations) dftotal = N - 1 Check yourself dftotal = Dfbetween + dfwithin

Need to first find dfbetween and dfwithin Dfbetween = (k = number of groups) dfwithin = (N = total number of observations) dftotal = 14 Check yourself 14 =

Look up F-critical using table F on pages F (2,12) = 3.88

Step 3: Calculate the F-value
Has 4 Sub-Steps 3.1) Calculate the needed ingredients 3.2) Calculate the SS 3.3) Calculate the MS 3.4) Calculate the F-value

Step 3.1: Ingredients X X2 Tj2 N n

Step 3.1: Ingredients

X X = 85 Xp = 40 Xn = 25 Xc = 20

X2 X = 85 X2 = 555 Xp = 40 Xn = 25 Xc = 20 X2n = 135 X2c = 90

T2 = (X)2 for each group X = 85 X2 = 555 Xp = 40 Xn = 25 Xc = 20
T2n = 625 T2c = 400 T2p = 1600

Tj2 X = 85 X2 = 555 Tj2 = 2625 Xp = 40 Xn = 25 Xc = 20
T2n = 625 T2c = 400 T2p = 1600

N X = 85 X2 = 555 Tj2 = 2625 N = 15 Xp = 40 Xn = 25 Xc = 20
T2n = 625 T2c = 400 T2p = 1600

n X = 85 X2 = 555 Tj2 = 2625 N = 15 n = 5 Xp = 40 Xn = 25
Xc = 20 X2n = 135 X2c = 90 X2p = 330 T2n = 625 T2c = 400 T2p = 1600

Step 3.2: Calculate SS SStotal X = 85 X2 = 555 Tj2 = 2625 N = 15

Step 3.2: Calculate SS 85 73.33 555 15 SStotal X = 85 X2 = 555
Tj2 = 2625 N = 15 n = 5 Step 3.2: Calculate SS SStotal 85 73.33 555 15

Step 3.2: Calculate SS SSWithin X = 85 X2 = 555 Tj2 = 2625 N = 15

Step 3.2: Calculate SS 2625 30 555 5 SSWithin X = 85 X2 = 555
Tj2 = 2625 N = 15 n = 5 Step 3.2: Calculate SS SSWithin 2625 30 555 5

Step 3.2: Calculate SS SSBetween X = 85 X2 = 555 Tj2 = 2625 N = 15

Step 3.2: Calculate SS 43.33 2625 85 5 15 SSBetween X = 85 X2 = 555
Tj2 = 2625 N = 15 n = 5 Step 3.2: Calculate SS SSBetween 43.33 2625 85 5 15

Step 3.2: Calculate SS Check! SStotal = SSBetween + SSWithin

Step 3.2: Calculate SS Check! 73.33 =

Step 3.3: Calculate MS

Step 3.3: Calculate MS 43.33 21.67 2

Step 3.3: Calculate MS 30 2.5 12

Step 3.4: Calculate the F value

21.67 8.67 2.5

Step 4: Decision If F value > than F critical

Step 5: Create the Summary Table

Step 6: Put answer into words
Question: Is the type of feedback a person receives significantly (.05) related their self-esteem? H1: The three population means are not all equal The type of feedback a person receives is related to their self-esteem

Practice You are interested in comparing the performance of three models of cars. Random samples of five owners of each car were used. These owners were asked how many times their car had undergone major repairs in the last 2 years.

Results

Practice Is there a significant (.05) relationship between the model of car and repair records?

Step 1: State the Hypothesis
H1: The three population means are not all equal H0: V = F = G

Need to first find dfbetween and dfwithin Dfbetween = (k = number of groups) dfwithin = (N = total number of observations) dftotal = 14 Check yourself 14 =

Look up F-critical using table F on pages F (2,12) = 3.88

Step 3.1: Ingredients X = 60 X2 = 304 Tj2 = 1400 N = 15 n = 5

Step 3.2: Calculate SS SStotal X = 60 X2 = 304 Tj2 = 1400 N = 15

Step 3.2: Calculate SS 60 64 304 15 SStotal X = 60 X2 = 304
Tj2 = 1400 N = 15 n = 5 Step 3.2: Calculate SS SStotal 60 64 304 15

Step 3.2: Calculate SS SSWithin X = 60 X2 = 304 Tj2 = 1400 N = 15

Step 3.2: Calculate SS 1400 24 304 5 SSWithin X = 60 X2 = 304
Tj2 = 1400 N = 15 n = 5 Step 3.2: Calculate SS SSWithin 1400 24 304 5

Step 3.2: Calculate SS SSBetween X = 60 X2 = 304 Tj2 = 1400 N = 15

Step 3.2: Calculate SS 40 1400 60 5 15 SSBetween X = 60 X2 = 304
Tj2 = 1400 N = 15 n = 5 Step 3.2: Calculate SS SSBetween 40 1400 60 5 15

Step 3.2: Calculate SS Check! SStotal = SSBetween + SSWithin

Step 3.2: Calculate SS Check! 64 =

Step 3.3: Calculate MS

Step 3.3: Calculate MS 40 20 2

Step 3.3: Calculate MS 24 2 12

20 10 2

Step 4: Decision If F value > than F critical

Step 5: Create the Summary Table

Step 6: Put answer into words
Question: Is there a significant (.05) relationship between the model of car and repair records? H1: The three population means are not all equal There is a significant relationship between the type of car a person drives and how often the car is repaired

A way to think about ANOVA
Make no assumption about Ho The populations the data may or may not have equal means

The samples can be used to estimate the variance of the population Assume that the populations the data are from have the same variance It is possible to use the same variances to estimate the variance of the populations

S2 = .50 S2 = .50 S2 = 5.0

Assume about Ho is true The population mean are not different from each other They are three samples from the same population All have the same variance and the same mean

2,1,2,3,2,5,4,3,4,4,9,6,3,7,5

Central Limit Theorem For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution a N (sample size) get larger. Furthermore, the sampling distribution of the mean will have a mean equal to  and a standard deviation equal to / N

Central Limit Theorem (remember) The variance of the means drawn from the same population equals the variance of the population divided by the sample size.

Can estimate population variance from the sample means with the formula *This only works if the means are from the same population

S2 = 4.00

*Estimates population variance only if the three means are from the same population

*Estimates population variance regardless if the three means are from the same population

What do all of these numbers mean?

Why do we call it “sum of squares”?
SStotal SSbetween SSwithin Sum of squares is the sum the squared deviations about the mean

Why do we use “sum of squares”?
SS are additive Variances and MS are only additive if df are the same

Another way to think about ANOVA
Think in “sums of squares” Represents the SS of all observations, regardless of the treatment.

Overall Mean= 4

Think in “sums of squares” Represents the SS deviations of the treatment means around the grand mean Its multiplied by n to give an estimate of the population variance (Central limit theorem)

Overall Mean= 4

Think in “sums of squares” Represents the SS deviations of the observations within each group

Overall Mean= 4

Sum of Squares SStotal SSbetween SSwithin
The total deviation in the observed scores SSbetween The total deviation in the scores caused by the grouping variable and error SSwithin The total deviation in the scores not caused by the grouping variable (error)

What if. . . You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days.

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What if. . . You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days.

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