New Statistical Tests, continued

1
New Statistical Tests, continued

• Analysis of Variance and F Test

2
Analysis of Variance

• The T test allows us to test whether two means
  are statistically significantly different.
• We now want to ask if we can test whether the
  means of more than two groups are statistically
  significantly different from one another.
• We are testing a relationship between a
  categorical independent variable and an interval
  dependent variable.

3
An Example

• Do the responses of students in the 1973
  Premarital Sexuality Survey differ by the year in
  school of the student?
• Do the students estimates of the proportion of
  female students having sexual intercourse differ
  by whether they are freshmen, sophomores,
  juniors, or seniors?

4
The Data

• The mean response for all the students was 47.5.
• The response by year in school was
• 1 42.1
• 2 45.9
• 3 50.1
• 4 52.0

5
The Data

• The mean response for all the students was 47.5.
• The response by year in school was
• 1 42.1
• 2 45.9
• 3 50.1
• 4 52.0
• Null hypothesis The differences observed are
  the result of chance.

6
More Data VisualizationDensity Display and
Error Bars
7
Concepts

• What is the overall variation in the responses?
  
• We calculate an overall mean response and a
  measure of dispersion, standard deviation and
  variance.
• We can calculate a mean response and a measure of
  dispersion, standard deviation and variance for
  each group.

8
Another Example

• From the data collected by Roger Simon on
  Milwaukee in 1905, are their differences in the
  mean number of people per family according to the
  type of building the family lives in., e.g., a
  cottage, a duplex, or a residence.

9
Basic Statistics for Firstfam
10
Basic statistics by Building Type
11
Overall distribution of Firstfam
12
Firstfam for Building Type Cottage
13
Firstfam for Building Type Residence
14
Firstfam for Building Type Duplex
15
Concepts

• We partition the total variation or variance into
  two components
• (1) variance which is a function of the group
  membership, that is the differences between the
  groups and
• (2) variance within the groups.
• More formally Total Sum of Squares Between
  Groups Sum of Squares Within Groups Sum of
  Squares

16
Equation

• Total Sum of Squares Within Groups Sum of
  Squares Between Groups Sum of Squares
• TSS SSW SSB

17
Calculations
18
Calculations
Case number VAR00001 GPMEAN GRANDMN
VARIANCE N NMINUS1 SSW
SSB 1 cottage 5.332
5.030 4.969 277.000 276.000 1371.444
25.264 2 duplex 4.410
5.030 3.537 83.000 82.000
290.034 31.957 3 residenc
4.842 5.030 4.628 171.000 170.000
786.760 6.044 4 total
5.030 5.030 4.739 531.000 .
. .
19
Degrees of Freedom

• DF between k -1
• DF within N k
• Website for F Table
• http//www.itl.nist.gov/div898/handbook/eda/sectio
  n3/eda3673.htmONE-05-1-10

20
SPSS Output

• Sum of Squares from previous slide
• Degrees of Freedom k-1 and N-k
• Mean Square Sum of Squares/df
• F Mean Square Between/Mean Square Within

21
Website for F Table

• http//www.itl.nist.gov/div898/handbook/eda/sectio
  n3/eda3673.htmONE-05-1-10

22
Strength of the Relationship

• Since Total Sum of Squares Within Groups Sum of
  Squares Between Groups Sum of Squares or TSS
  SSW SSB.
• Between Groups Sum of Squares/Total Sum of
  Squares Proportion of Variance Explained or Eta
  Squared
• SSB/TSS Eta Squared
• Eta Squared is equivalent to R Squared