Analysis of Variance

ANOVA (Analysis of Variance)

ANOVA na naman toh!!!!!!!!!!!!!!!!!!!!

Reality Human beings are complex beings. Our

interaction is also complex.

Not every human phenomenon/ social reality can be conveniently sliced in two groups.

Social researchers often seeks to compare more than two groups or samples.

T-test as an statistical method is limited in this sense.

It is a lot of work!

Increasing risk (probability) to commit Type I Error (yielding statistically significant finding due to sampling error rather than by the true population difference.

Need for a single overall decision.

Analysis of Variance alyas ANOVA

Why not do series of t-tests?

The Logic of Analysis of Variance

Three aspects of variation

Total variation

Variation within groups

Variation between groups

Instead of t ratio, ANOVA yields F ratio (variation

between groups and variation within groups are compared).

Hypotheses:

Ho: 1 = 2 = 3 = j

Ha: i j

Copyright Pearson Education, Inc., Allyn & Bacon 2010

Variation Within Groups/Residuals

Variation Between Groups/ Model



The Sum of Squares

Sum of squares - the heart of ANOVA

Sum of squared deviations from the mean

Total (SStotal), Between (SSbetween/model), Within (SSwithin/residual)

Formulas:

2)( totaltotal XXSS

2)( groupwithin XXSS

2)( totalgroupgroupbetween XXNSS

22

22

22

totaltotalgroupgroupbetween

totalgrouptotalwithin

totaltotaltotaltotal

XNXNSS

XNXSS

XNXSS

These are much simpler formulas:


Mean Square (note: estimates of variance that exist in the population)

Value of sums of squares grows as variation increases

Value also increases with sample size

Mean square (or variance) control for these

influences

Formulas:

within

withinwithin

between

betweenbetween

df

SSMS

df

SSMS

kNdf

kdf

totalwithin

between

1

Where k = number of groups


The F Ratio

ANOVA yields an F ratio

Variation between and within groups are compared

F ratio must be evaluated for significance

The larger the F ratio, the more likely it is to be

statistically significant (F ratio > F critical)

Must calculate degrees of freedom (within

[denominator] and between [numerator])

Interpret F ratio with Table D in the appendix

within

between

MS

MSF


A Multiple Comparison of Means

A significant F ratio tells us there is a difference among groups.

If we had only two samples, no other test would be necessary.

With three or more samples, we need to determine where the difference lies.

We do this with Tukeys HSD (Honestly Significant Difference).


Tukeys HSD (note: a post hoc test) Tukeys HSD (Honestly Significant Difference) only makes

sense after a significant F ratio has been identified.

This allows for a comparison of any two means against Tukeys HSD calculation.

Takes into account Type I errors

group

within

N

MSqHSD

q = table value at a given level of significance for the total number of group means being compared MSwithin = within-groups mean square (obtained from the analysis of variance) Ngroup = number of subjects in each group (assumes the same number in each group)


Summary

t ratio limited F ratio allows comparison of two or more means

F ratio significance is interpreted using Table D ( F critical).

Tukeys HSD isolates significant differences


Requirements for Using the F Ratio

1. A comparison between two or more

independent means

2. Data at the interval level of measurement

3. Random sampling techniques

4. A normal distribution

5. Equal variances assumed

Example of ANOVA (One-way)

nce upon a time in a far far away land, there lived

a witch who is so obsessed of K-pop and TRUE LOVE. To make the story short, she concocted a LOVE potion out of Cupids urine, Austin Powers mojo, mermaids tongues, Taylor Swifts teardrops on her guitar, sugar and spice, and everything niceand also not to forget chemical X blah blah blah blah blah blah blah blah blah blah blah the witch became so filthy rich. THE END

The Love Potion

As Merlins experimental researcher, you are interested about the effect of different concentration of this LOVE potion to the lovers lovelovelove. Given the lovelovelove levels(1-10) of the three sample groups (Placebo group, Low

dose, and High dose); you want to predict the levels of lovelovelove from different levels (dosage) of Love potion. You may use 95% confidence interval.

Placebo group

3

2

1

1

4

Low Dose group

5

2

4

2

3

High Dose group

7

4

5

3

6

Hypotheses

Ho: There is no significant difference of lovelovelove levels of participants when grouped according to different dosage of the Love potion. Thus the potion has no effect in inducing lovelovelove.

Ha: There is a significant difference of lovelovelove levels of participants when grouped according to different dosage of the Love potion. Thus the potion has effect in inducing lovelovelove.

Placebo Group Low Dose Group High Dose Group

(N1 = 5) (N2 = 5) (N3 = 5)

X1 X12 X2 X2

2 X3 X32

3 9 5 25 7 49

2 4 2 4 4 16

1 1 4 16 5 25

1 1 2 4 3 9

4 16 3 9 6 36

X1 = 11 X12 = 31 X2 = 16 X2

2 = 58 X3 = 25 X32 = 135

X1 = 2.2 X2 = 3.2 X3 = 5.0

Step 1: Find the mean for each sample

X = X/N

X1 = 2.2

X2 = 3.2

X3 = 5.0

Step 2: Find the sum of scores, sum of squared scores, number of subjects, and mean for all groups combined.

Xtotal = X1 + X2 + X3

= 11 + 16 + 25

= 52

Step 2 continuation X2total = X1

2+ X22+ X3

2

= 31 + 58 + 135

= 224

Ntotal = N1 + N2 + N3 = 5+ 5 + 5

= 15

Xtotal = Xtotal / Ntotal = 52/15

= 3.47

Step 3: Find the total sum of squares. SStotal = X

2total - Ntotal X

2total

= 224 (15) (3.47)2 = 224 (15) (12.04)

= 224 180.6

= 43.4

Step 4: Find the within-groups sum of squares SSwithin = X

2total - Ngroup X

2group

= 224 [ (5) (2.2)2 + (5) (3.2)2 + (5) (5.0)2 ]

= 224 [ (5) (4.84) + (5) (10.24) + (5) (25) ]

= 224 (24.2 + 51.2 + 125)

= 224 200.4

= 23.6

Step 5: Find the between-groups sum of squares. SSbetween = Ngroup X

2group - Ntotal X

2total

= [ (5) (2.2)2 + (5) (3.2)2 + (5) (5.0)2 ] (15) (3.47)2

= [ (5) (4.84) + (5) (10.24) + (5) (25) ] (15) (12.04)

= (24.2 + 51.2 + 125) - 180.6

= 200.4 180.6

= 19.8

Step 6: Find the between-groups degrees of freedom. dfbetween = k 1

= 3 1

= 2

Step 7: Find the within-groups degrees of freedom.

dfwithin = Ntotal k

= 15 3

= 12

Step 8: Find the within-groups mean square.

MSwithin = SSwithin/ dfwithin = 23.6/12

= 1.97

Step 9: Find the between-groups mean square.

MSbetween = SSbetween/dfbetween = 19.8/2

= 9.9

Step 10: Obtain the F-ratio.

F = MSbetween/MSwithin =9.9/1.97

=5.025

Step 11: Compare F-obtained with F-table.

F-obtained > F-table

F-obtained = 5.025

F-table (3.88, to other references 3.89)

df = df between/ df within 2 (column) and 12 (row)

= 0.05

5.025 > 3.88, TRUE!!!! Reject Ho!

E-celebrate na yan!

F-Distribution table used: Levin, Fox, and Forde, 2013)

Mag-manwal Kalkyulesyon napud ta Sir, foreves?!!!!!

Thanks to Merlin, he uses SPSS!

Lets go analyze this LOVEly problem with SPSS: using the Love Potion data set

Descriptives

LoveLoveLove N Mean Std.

Deviation Std. Error

95% Confidence Interval for Mean

Minimum Maximum

Lower Bound Upper Bound

Placebo 5 2.2000 1.30384 .58310 .5811 3.8189 1.00 4.00

Low Dose 5 3.2000 1.30384 .58310 1.5811 4.8189 2.00 5.00

High Dose 5 5.0000 1.58114 .70711 3.0368 6.9632 3.00 7.00

Total 15 3.4667 1.76743 .45635 2.4879 4.4454 1.00 7.00

ANOVA

LoveLoveLove Sum of Squares df Mean Square F Sig.

Between Groups 20.133 2 10.067 5.119 .025

Within Groups 23.600 12 1.967

Total 43.733 14

Multiple Comparisons

Dependent Variable: LoveLoveLove

Tukey HSD (I) Dosage of Love Potion

(J) Dosage of Love Potion

Mean Difference

(I-J)

Std. Error

Sig. 95% Confidence Interval

Lower Bound

Upper Bound

Placebo Low Dose

-1.00000 .88694 .516 -3.3662 1.3662

High Dose -2.80000* .88694 .021 -5.1662 -.4338

Low Dose Placebo

1.00000 .88694 .516 -1.3662 3.3662

High Dose -1.80000 .88694 .147 -4.1662 .5662

High Dose Placebo

2.80000* .88694 .021 .4338 5.1662

Low Dose 1.80000 .88694 .147 -.5662 4.1662

*. The mean difference is significant at the 0.05 level.

Decision for our Null Hypothesis

Reject Ho

(P

How about the effect size?

Eta squared (2) a.k.a R2

Effect size

Omega squared

Thank you wizard trainees

for listening

Analysis of Variance

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