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 H 1: The three population means are not all equal H 0 :  pos =  neg =  cont

Step 2: Find F-Critical Step 2.1 Need to first find df between and df within Df between = k - 1 (k = number of groups) df within = N - k (N = total number of observations) df total = N - 1 Check yourself df total = Df between + df within

Step 2: Find F-Critical Step 2.1 Need to first find df between and df within Df between = 2 (k = number of groups) df within = 12 (N = total number of observations) df total = 14 Check yourself 14 =

Step 2: Find F-Critical Step 2.2 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  X 2  T j 2 N n

Step 3.1: Ingredients

XX  X p = 40  X n = 25  X c = 20  X = 85

X2X2  X p = 40  X n = 25  X c = 20  X = 85  X 2 = 555  X 2 p = 330  X 2 n = 135  X 2 c = 90

T 2 = (  X) 2 for each group  X p = 40  X n = 25  X c = 20  X = 85  X 2 = 555  X 2 p = 330  X 2 n = 135  X 2 c = 90 T 2 p = 1600 T 2 n = 625T 2 c = 400

Tj2Tj2  X p = 40  X n = 25  X c = 20  X = 85  X 2 = 555  T j 2 = 2625  X 2 p = 330  X 2 n = 135  X 2 c = 90 T 2 p = 1600 T 2 n = 625T 2 c = 400

N  X p = 40  X n = 25  X c = 20  X = 85  X 2 = 555  T j 2 = 2625 N = 15  X 2 p = 330  X 2 n = 135  X 2 c = 90 T 2 p = 1600 T 2 n = 625T 2 c = 400

n  X p = 40  X n = 25  X c = 20  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5  X 2 p = 330  X 2 n = 135  X 2 c = 90 T 2 p = 1600 T 2 n = 625T 2 c = 400

Step 3.2: Calculate SS  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5 SS total

Step 3.2: Calculate SS SS total  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5

Step 3.2: Calculate SS SS Within  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5

Step 3.2: Calculate SS SS Within  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5

Step 3.2: Calculate SS SS Between  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5

Step 3.2: Calculate SS SS Between  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5

Step 3.2: Calculate SS Check! SS total = SS Between + SS Within

Step 3.2: Calculate SS Check! =

Step 3.3: Calculate MS

Calculating this Variance Ratio

Step 3.3: Calculate MS

Step 3.4: Calculate the F value

Step 3.4: Calculate the F value

Step 4: Decision If F value > than F critical –Reject H 0, and accept H 1 If F value < or = to F critical –Fail to reject H 0

Step 4: Decision If F value > than F critical –Reject H 0, and accept H 1 If F value < or = to F critical –Fail to reject H 0 F value = 8.67 F crit = 3.88

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? H 1: The three population means are not all equal The type of feedback a person receives is related to their self-esteem

SPSS

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 H 1: The three population means are not all equal H 0 :  V =  F =  G

Step 2: Find F-Critical Step 2.1 Need to first find df between and df within Df between = 2 (k = number of groups) df within = 12 (N = total number of observations) df total = 14 Check yourself 14 =

Step 2: Find F-Critical Step 2.2 Look up F-critical using table F on pages F (2,12) = 3.88

Step 3.1: Ingredients  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

Step 3.2: Calculate SS  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5 SS total

Step 3.2: Calculate SS SS total  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

Step 3.2: Calculate SS SS Within  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

Step 3.2: Calculate SS SS Within  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

Step 3.2: Calculate SS SS Between  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

Step 3.2: Calculate SS SS Between  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

Step 3.2: Calculate SS Check! SS total = SS Between + SS Within

Step 3.2: Calculate SS Check! 64 =

Step 3.3: Calculate MS

Calculating this Variance Ratio

Step 3.3: Calculate MS

Step 3.4: Calculate the F value

Step 3.4: Calculate the F value

Step 4: Decision If F value > than F critical –Reject H 0, and accept H 1 If F value < or = to F critical –Fail to reject H 0

Step 4: Decision If F value > than F critical –Reject H 0, and accept H 1 If F value < or = to F critical –Fail to reject H 0 F value = 10 F crit = 3.88

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? H 1: 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 H o –The populations the data may or may not have equal means

A way to think about ANOVA 2 4 6

A way to think about ANOVA 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

S 2 =.50 S 2 = 5.0

A way to think about ANOVA

Assume about H o 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

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

A way to think about ANOVA 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

A way to think about ANOVA 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

A way to think about ANOVA 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.

A way to think about ANOVA Can estimate population variance from the sample means with the formula *This only works if the means are from the same population

A way to think about ANOVA S 2 = 4.00

A way to think about ANOVA

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

A way to think about ANOVA *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”? SS total SS between SS within 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.

Another way to think about ANOVA Overall Mean= 4

Another way to think about ANOVA

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=

Another way to think about ANOVA

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

Overall Mean=

Another way to think about ANOVA

Sum of Squares SS total –The total deviation in the observed scores SS between –The total deviation in the scores caused by the grouping variable and error SS within –The total deviation in the scores not caused by the grouping variable (error)

Conceptual Understanding Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at.05.

Conceptual Understanding Fcrit = 3.18 Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at.05. Fcrit (2, 57) = 3.15

Conceptual Understanding Distinguish between: Between-group variability and within-group variability

Conceptual Understanding Distinguish between: Between-group variability and within-group variability Between concerns the differences between the mean scores in various groups Within concerns the variability of scores within each group

Between and Within Group Variability Between-group variability Within-group variability

Between and Within Group Variability sampling error + effect of variable sampling error

Conceptual Understanding Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00?

Conceptual Understanding Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00? F ratio will approach 1.00 when the null hypothesis is true F ratio will be greater than 1.00 when the null hypothesis is not true

Conceptual Understanding Without computing the SS within, what must its value be? Why?

Conceptual Understanding The SS within is 0. All the scores within a group are the same (i.e., there is NO variability within groups)

Example Freshman, Sophomore, Junior, Senior Measure Happiness (1-100)

ANOVA Traditional F test just tells you not all the means are equal Does not tell you which means are different from other means

Why not Do t-tests for all pairs Fresh vs. Sophomore Fresh vs. Junior Fresh vs. Senior Sophomore vs. Junior Sophomore vs. Senior Junior vs. Senior

Problem What if there were more than four groups? Probability of a Type 1 error increases. Maximum value = comparisons (.05) 6 (.05) =.30

Chapter 12 A Priori and Post Hoc Comparisons Multiple t-tests Linear Contrasts Orthogonal Contrasts Trend Analysis Bonferroni t Fisher Least Significance Difference Studentized Range Statistic Dunnett’s Test

Multiple t-tests Good if you have just a couple of planned comparisons Do a normal t-test, but use the other groups to help estimate your error term Helps increase you df

Remember

Note

Proof CandyGender

t = /.641 = 4.16

Also, when F has 1 df between

Within Variability Within variability of all the groups represents “error” You can therefore get a better estimate of error by using all of the groups in your ANOVA when computing a t-value

Note: This formula is for equal n

Hyp 1: Juniors and Seniors will have different levels of happiness Hyp 2: Seniors and Freshman will have different levels of happiness

Hyp 1: Juniors and Seniors will have different levels of happiness

t crit (20 df) = 2.086

Hyp 1: Juniors and Seniors will have different levels of happiness t crit (20 df) = Juniors and seniors do have significantly different levels of happiness

Hyp 2: Seniors and Freshman will have different levels of happiness

t crit (20 df) = 2.086

Hyp 2: Seniors and Freshman will have different levels of happiness t crit (20 df) = Freshman and seniors do not have significantly different levels of happiness

Hyp 1: Juniors and Sophomores will have different levels of happiness Hyp 2: Seniors and Sophomores will have different levels of happiness PRACTICE!

Practice To investigate the maternal behavior of lab rats, we move the rat pup a fixed distance from the mother and record the time required for the mother to retrieve the pup. We run the study with 5, 20, and 35 day old pups. Figure out if 5 days is different than 35 days. SPSS Homework (Do the ANOVA analysis in SPSS – use output to answer question above) 5 days days days