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One-Way Analysis of Variance

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One-Way Analysis of Variance to compare 2 or population means Does distance it takes to stop car at 60 mph depend on tire brand? Comparison of Five Tire Brands ... – PowerPoint PPT presentation

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Title: One-Way Analysis of Variance


1
One-Way Analysis of Variance
  • to compare 2 or population means

2
Does distance it takes to stop car at 60 mph
depend on tire brand?
Brand1 Brand2 Brand3 Brand4 Brand5 194
189 185 183 195 184 204 183 193 197
189 190 186 184 194 189 190 183
186 202 188 189 179 194 200 186
207 191 199 211 195 203 188 196 203
186 193 196 188 206 183 181 189
193 202 188 206 194 196 195
3
Comparison of Five Tire BrandsStopping Distance
at 60 mph
4
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
5
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

6
Analysis of Variance
Analysis of Variance for comparing all 5
brands Source DF SS MS F
P 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.
7
Does learning method affect students 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.

8
Suppose Study 1
Is there a reasonable conclusion?
9
Suppose Study 2
Is there a reasonable conclusion?
10
Suppose Study 3
Is there a reasonable conclusion?
11
Analysis of Variance
The variation between the group means and the
grand mean is larger than the variation within
each of the groups.
12
Analysis of Variance
The variation between the group means and the
grand mean is smaller than the variation within
each of the groups.
13
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.

14
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 find the components of variation
in this column
DF means degrees of freedom
SS means sums of squares
MS means mean squared
15
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 of interest
Error means Variability within groups or
unexplained random variation
Total means Total variation from the grand
mean
16
ANOVA Notation
  • N number of total data values.
  • ni number of data values in ith group
  • t number of groups (or factor levels)
  • yij jth data value in ith group
  • Yi. sum of data values in ith group
  • Y.. sum of all data values

17
General ANOVA Table
One-way Analysis of Variance Source DF
SS MS F P Factor t-1
SS(Between) MSB MSB/MSE Error N-t SS(Error)
MSE Total N-1 SS(Total)
From F-distribution with t-1 numerator and N-t
denominator d.f.
MSB SS(Between)/(t-1) MSE SS(Error)/(N-t)
N-1 (t-1) (N-t)
SS(Total) SS(Between) SS(Error)
18
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
1255.2 2510.5/2 13.4 161.2/12
14 2 12
93.44 1255.3/13.4
2671.7 2510.5 161.2
19
Recall Study 3
20
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 pretty large so cannot reject the
null hypothesis. There is insufficient evidence
to conclude that the average exam scores differ
for the three learning methods.
21
One-Way ANOVA with Unstacked Data
DATA
IN MINITAB
std1 osm1 shk1 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.
22
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.
23
Do Holocaust survivors have more sleep problems
than others?
24
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
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