One factor analysis of variance
This presentation is the property of its rightful owner.
Sponsored Links
1 / 28

One-Factor Analysis of Variance PowerPoint PPT Presentation


  • 73 Views
  • Uploaded on
  • Presentation posted in: General

One-Factor Analysis of Variance. A method to compare two or more (normal) population means. Does distance it takes to stop car at 60 mph depend on tire brand ?. Brand1 Brand2 Brand3 Brand4 Brand5 194 189185 183 195 184 204183 193 197 189 190186 184 194

Download Presentation

One-Factor Analysis of Variance

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


One factor analysis of variance

One-Factor Analysis of Variance

A method to compare two or more (normal) population means


Does distance it takes to stop car at 60 mph depend on tire brand

Does distance it takes to stop car at 60 mph depend on tire brand?

  • Brand1 Brand2 Brand3 Brand4 Brand5

  • 194 189185 183 195

  • 184 204183 193 197

  • 189 190186 184 194

  • 189 190183 186 202

  • 188 189179 194 200

  • 186 207191 199 211

  • 195 203188 196 203

  • 186 193196 188 206

  • 183 181189 193 202

  • 188 206194 196 195


Comparison of five tire brands stopping distance at 60 mph

Comparison of five tire brands(stopping distance at 60 mph)


Sample descriptive statistics

Sample descriptive statistics

Brand N MEAN SD

1 10 188.20 3.88

2 10 195.20 9.02

3 10 187.40 5.27

4 10 191.20 5.55

5 10 200.50 5.44


Hypotheses

Hypotheses

  • The null hypothesis is that the group population means are all the same. That is:

    • H0: 1 = 2 = 3 = 4 = 5

  • The alternative hypothesis is that at least one group population mean differs from the others. That is:

    • HA: at least one i differs from the others


Analysis of variance

Analysis of Variance

Analysis of Variance

for comparing all 5 brands

Source DF SS MS FP

Brand 4 1174.8 293.7 7.95 0.000

Error 45 1661.7 36.9

Total 49 2836.5

The P-value is small (0.000, to three decimal places), so reject the null hypothesis. There is sufficient evidence to conclude that at least one brand is different from the others.


Does learning method affect student s exam scores

Does learning method affect student’s exam scores?

  • Consider 3 methods:

    • standard

    • osmosis

    • shock therapy

  • Convince 15 students to take part. Assign 5 students randomly to each method.

  • Wait eight weeks. Then, test students to get exam scores.


Suppose study 1

Suppose … Study #1

Is there a reasonable conclusion?


Suppose study 2

Suppose … Study #2

Is there a reasonable conclusion?


Suppose study 3

Suppose … Study #3

Is there a reasonable conclusion?


Analysis of variance1

“Analysis of Variance”

The variation between the group means and the grand mean is larger than the variation within the groups.


Analysis of variance2

“Analysis of Variance”

The variation between the group means and the grand mean is smaller than the variation within the groups.


Analysis of variance3

Analysis of Variance

  • A division of the overall variability in data values in order to compare means.

  • Overall (or “total”) variability is divided into two components:

    • the variability “between” groups, and

    • the variability “within” groups

  • Summarized in an “ANOVA” table.


Anova table for study 1

ANOVA Table for Study #1

“F” means “F test statistic”

One-way Analysis of Variance

Source DF SS MS F P

Factor 2 2510.5 1255.3 93.44 0.000

Error 12 161.2 13.4

Total 14 2671.7

P-Value

“Source” means “the source of the variation in the data”

“DF” means “the degrees of freedom”

“SS” means “the sum of squares”

“MS” means “mean sum of squares”


Anova table for study 11

ANOVA Table for Study #1

One-way Analysis of Variance

Source DF SS MS F P

Factor 2 2510.5 1255.3 93.44 0.000

Error 12 161.2 13.4

Total 14 2671.7

“Factor” means “Variability between groups” or “Variability due to the factor (or treatment) of interest”

“Error” means “Variability within groups” or “unexplained random error”

“Total” means “Total variation from the grand mean”


Anova notation

ANOVA Notation


General anova table

General ANOVA Table

One-way Analysis of Variance

Source DF SS MS F P

Factor m-1 SS(Between) MSB MSB/MSE

Error n-m SS(Error) MSE

Total n-1 SS(Total)

From F-distribution with m-1 numerator and n-m denominator d.f.

MSB = SS(Between)/(m-1)

MSE = SS(Error)/(n-m)

n-1 = (m-1) + (n-m)

SS(Total) = SS(Between) + SS(Error)


Anova table for study 12

ANOVA Table for Study #1

One-way Analysis of Variance

Source DF SS MS F P

Factor 2 2510.5 1255.3 93.44 0.000

Error 12 161.2 13.4

Total 14 2671.7

1255.3 = 2510.5/2

13.4 = 161.2/12

14 = 2 + 12

93.44 = 1255.3/13.4

2671.7 = 2510.5 + 161.2


Total sum of squares ss to

Total sum of squares SS(TO)

Definition:

Shortcut:


Treatment sum of squares ss t

Treatment sum of squares SS(T)

Definition:

Shortcut:


Error sum of squares ss e

Error sum of squares SS(E)

Definition:

Shortcut:


Ss to ss t ss e

SS(TO) = SS(T) + SS(E)

We’ve broken down the TOTAL variation into a component due to TREATMENT and a component due to random ERROR.


Recall study 3

Recall Study #3


Anova table for study 3

ANOVA Table for Study #3

One-way Analysis of Variance

Source DF SS MS F P

Factor 2 80.1 40.1 0.46 0.643

Error 12 1050.8 87.6

Total 14 1130.9

The P-value is large so we cannot reject the null hypothesis. There is insufficient evidence to conclude that the average exam scores differ for the three learning methods.


One way anova with unstacked data

One-Way ANOVA with Unstacked Data

DATA:

IN MINITAB:

std1osm1shk1

51 58 77

45 68 72

40 64 78

41 63 73

41 62 75

1. Select Stat.

2. Select ANOVA.

3. Select One-way (Unstacked).

4. Select the columns containing the data.

5. If you want boxplots or dotplots of the data, select Graphs...

6. Select OK.


One way anova with stacked data

One-Way ANOVA with Stacked Data

DATA:

Method Score

1 51

1 45

1 40

1 41

1 41

2 58

2 68

2 64

2 63

2 62

3 77

3 72

3 78

3 73

3 75

IN MINITAB:

1. Select Stat.

2. Select ANOVA.

3. Select One-way.

4. Select the “response.” (Score)

5. Select the “factor.” (Method)

5. If you want boxplots or dotplots of the data, select Graphs...

6. Select OK.


Do holocaust survivors have more sleep problems than others

Do Holocaust survivors have more sleep problems than others?


Anova table for sleep study

ANOVA Table for Sleep Study

One-way Analysis of Variance

Source DF SS MS F P

Factor 2 1723.8 861.9 61.69 0.000

Error 117 1634.8 14.0

Total 119 3358.6


  • Login