Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.
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Transcript of Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.
![Page 1: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/1.jpg)
Chapter 11
HYPOTHESIS TESTING USING THEONE-WAY ANALYSIS OF VARIANCE
![Page 2: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/2.jpg)
Moving Forward
Your goals in this chapter are to learn:• The terminology of analysis of variance• When and how to compute • Why should equal 1 if H0 is true, and why
it is greater than 1 if H0 is false
• When and how to compute Tukey’s HSD• How eta squared describes effect size
obtF
obtF
![Page 3: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/3.jpg)
Analysis of Variance
• The analysis of variance is the parametric procedure for determining whether significant differences occur in an experiment with two or more sample means
• In an experiment involving only two conditions of the independent variable, you may use either a t-test or the ANOVA
![Page 4: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/4.jpg)
An Overview of ANOVA
![Page 5: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/5.jpg)
One-Way ANOVA
• Analysis of variance is abbreviated as ANOVA• An independent variable is also called a factor• Each condition of the independent variable is
called a level or treatment• Differences produced by the independent
variable are a treatment effect
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Between-Subjects
• A one-way ANOVA is performed when one independent variable is tested in the experiment
• When an independent variable is studied using independent samples in all conditions, it is called a between-subjects factor
• A between-subjects factor involves using the formulas for a between-subjects ANOVA
![Page 7: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/7.jpg)
Within-Subjects Factor
• When a factor is studied using related (dependent) samples in all levels, it is called a within-subjects factor
• This involves a set of formulas called a within-subjects ANOVA
![Page 8: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/8.jpg)
Diagram of a Study Having ThreeLevels of One Factor
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Assumptions of the ANOVA
1. All conditions contain independent samples
2. The dependent scores are normally distributed, interval or ratio scores
3. The variances of the populations are homogeneous
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Experiment-Wise Error
• The probability of making a Type I error somewhere among the comparisons in an experiment is called the experiment-wise error rate
• When we use a t-test to compare only two means in an experiment, the experiment-wise error rate equals
![Page 11: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/11.jpg)
Comparing Means
• When there are more than two means in an experiment, the multiple t-tests result in an experiment-wise error rate much larger than the we have selected
• Using the ANOVA allows us to make all our decisions and keep the experiment-wise error rate equal to
![Page 12: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/12.jpg)
Statistical Hypotheses
kH 210 :
equalaresallnot:a H
![Page 13: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/13.jpg)
The F-Test
• The statistic for the ANOVA is F
• When Fobt is significant, it indicates only that somewhere among the means at least two of them differ significantly
• It does NOT indicate which specific means differ significantly
• When the F-test is significant, we perform post hoc comparisons
![Page 14: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/14.jpg)
Post Hoc Comparisons
• Post hoc comparisons are like t-tests
• We compare all possible pairs of level means from a factor, one pair at a time to determine which means differ significantly from each other
![Page 15: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/15.jpg)
Components of the ANOVA
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Mean Squares
• The mean square within groups describes the variability in scores within the conditions of an experiment. It is symbolized by MSwn.
• The mean square between groups describes the differences between the means of the conditions in a factor. It is symbolized by MSbn.
![Page 17: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/17.jpg)
The F-Ratio
• The F-ratio equals the mean square between groups divided by the mean square within groups
• When H0 is true, Fobt should equal 1
• When H0 is false, Fobt should be greater than 1
wn
bnobt MS
MSF
![Page 18: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/18.jpg)
Performing the ANOVA
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Sum of Squares
• The computations for the ANOVA require the use of several sums of squared deviations
• The sum of squares is the sum of the squared deviations of a set of scores around the mean of those scores
• It is symbolized by SS
![Page 20: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/20.jpg)
Summary Table of a One-way ANOVA
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Computing Fobt
1. Compute the sums and means• • •
for each level. Add the from all levels to get . Add together the from all levels to get . Add the ns together to get N.
X2X
X
X
totX 2X2totX
![Page 22: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/22.jpg)
Computing Fobt
2. Compute the total sum of squares (SStot)
N
XXSS
2tot2
tottot
)(
![Page 23: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/23.jpg)
Computing Fobt
3. Compute the sum of squares between groups (SSbn)
N
X
n
XSS
2tot
2
bn
)(
columnin
)columnin(
![Page 24: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/24.jpg)
Computing Fobt
4. Compute the sum of squares within groups (SSwn)
bntotwn SSSSSS
![Page 25: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/25.jpg)
Computing Fobt
Compute the degrees of freedom• The degrees of freedom between groups
equals k – 1 where k is the number of levels in the factor
• The degrees of freedom within groups equals N – k
• The degrees of freedom total equals N – 1
![Page 26: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/26.jpg)
Computing Fobt
Compute the mean squares
•
•
bn
bnbn df
SSMS
wn
wnwn df
SSMS
![Page 27: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/27.jpg)
Computing Fobt
Compute Fobt
wn
bnobt MS
MSF
![Page 28: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/28.jpg)
Sampling Distribution of FWhen H0 Is True
![Page 29: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/29.jpg)
Degrees of Freedom
The critical value of F (Fcrit) depends on
• The degrees of freedom (both the dfbn = k – 1 and the dfwn = N – k)
• The selected
• The F-test is always a one-tailed test
![Page 30: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/30.jpg)
Tukey’s HSD Test
When the ns in all levels of the factor are equal, use the Tukey HSD multiple comparisons test
where qk is found using Table 5 in Appendix B
n
MSqHSD k
wn)(
![Page 31: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/31.jpg)
Tukey’s HSD Test
• Determine the difference between each pair of means
• Compare each difference between the means to the HSD
• If the absolute difference between two means is greater than the HSD, then these means differ significantly
![Page 32: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/32.jpg)
Effect Size and Eta2
![Page 33: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/33.jpg)
Proportion of Variance Accounted For
Eta squared indicates the proportion of variance in the dependent variable scores that is accounted for by changing the levels of a factor
)( 2
tot
bn2
SS
SS
![Page 34: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/34.jpg)
Example
Using the following data set, conduct a one-way ANOVA. Use = 0.05.
Group 1 Group 2 Group 3
14 14 10 13 11 15
13 10 12 11 14 13
14 15 11 10 14 15
![Page 35: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/35.jpg)
Example
611.5518
2292969
)(
2
2tot2
tottot
N
XXSS
![Page 36: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/36.jpg)
Example
111.2218
229
6
82
6
67
6
80
)(
columnin
)columnin(
2222
2tot
2
bn
N
X
n
XSS
![Page 37: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/37.jpg)
Example
50.33
111.22611.55bntotwn
SSSSSS
![Page 38: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/38.jpg)
Example
• dfbn = k – 1 = 3 – 1 = 2
• dfwn = N – k = 18 – 3 = 15
• dftot = N – 1 = 18 – 1 = 17
![Page 39: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/39.jpg)
Example
055.112
111.22
bn
bnbn
df
SSMS
233.215
50.33
wn
wnwn
df
SSMS
951.4233.2
055.11
wn
bnobt
MS
MSF
![Page 40: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/40.jpg)
Example
• Fcrit for 2 and 15 degrees of freedom and = 0.05 is 3.68
• Since Fobt = 4.951, the ANOVA is significant
• A post hoc test must now be performed
![Page 41: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/41.jpg)
Example
242.26
233.2675.3)( wn
n
MSqHSD k
334.0333.13667.13
500.2167.11667.13
166.2167.11333.13
13
23
21
XX
XX
XX
![Page 42: Chapter 11 HYPOTHESIS TESTING USING THE ONE-WAY ANALYSIS OF VARIANCE.](https://reader035.fdocuments.in/reader035/viewer/2022062217/56649e8f5503460f94b92b1b/html5/thumbnails/42.jpg)
Example
Because 2.50 > 2.242 (HSD), the mean of sample 3 is significantly different from the mean of sample 2.