Chapter 9: The Analysis of Variance (ANOVA)

http://www.luchsinger-mathematics.ch/Var_Reduction.jpg

ANOVA: Examples

1) Do four different brands of gasoline have different effects on automobile efficiency?

2) Does the type of sugar solution (glucose, sucrose, fructose, mixture) have an effect on bacterial growth?

3) Does the resulting color density of a fabric depend on the amount of dye used?

ANOVA: Graphical

ANOVA test statistic

ANOVA test

F Distribution

http://www.vosesoftware.com/ModelRiskHelp/index.htm#Distributions/Continuous_distributions/F_distribution.htm

P-value for an upper-tailed F test

shaded area=P-value = 0.05

Table IVF critical values

ANOVA Table: FormulasSource df SS MS

(Mean Square)F

Model(Between) k – 1

Error (Within) n – k

Total n – 1

k2

i ii 1

n (x x)

k

2i i

i 1

(n 1)s

ink

2ij

i 1 j 1

(x x)

SSM SSM

dfm k 1

SSE SSE

dfe n k

MSM

MSE

ANOVA Hypothesis test: Summary

H0: μ1 = μ2 = = μk

Ha: At least two μ’s are different

Test statistic: P-value: area under the upper tail with an

F distribution with (dfm, dfe) degrees of freedom

MSMF

MSE

ANOVA: Example

A random sample of 15 healthy young men are split randomly into 3 groups of 5. They receive 0, 20, and 40 mg of the drug Paxil for one week. Then their serotonin levels are measured to determine whether Paxil affects serotonin levels. The data is on the next slide.

Does Paxil affect serotonin levels in healthy young men?

ANOVA: Example (cont).Dose 0 mg 20 mg 40 mg

48.62 58.60 68.5949.85 72.52 78.2864.22 66.72 82.7762.81 80.12 76.5362.51 68.44 72.33

ANOVA: Example (cont)

1.Let 1 be the mean serotonin level for men receiving 0 mg of Paxil.Let 2 be the mean serotonin level for men receiving 20 mg of Paxil.Let 3 be the mean serotonin level for men receiving 40 mg of Paxil.

ANOVA: Example (cont)

2.H0: 1 = 2 = 3; mean serotonin levels are the same at all 3 dosage levels [or, mean serotonin levels are unaffected by Paxil dose]HA: The mean serotonin levels of the three groups are not all equal. [or, serotonin levels are affected by Paxil does]

ANOVA: Example (cont).Dose 0 mg 20 mg 40 mg

48.62 58.60 68.5949.85 72.52 78.2864.22 66.72 82.7762.81 80.12 76.5362.51 68.44 72.33 overall

ni 5 5 5 15xNi 57.60 69.28 75.70 67.53si 7.678 7.895 5.460(ni-1)si

2 235.78 249.32 119.24 604.34ni(xNi - xNN)2 492.56 15.36 333.96 841.88

ANOVA: Example (cont)Source df SS MS 4. F-Ratio 5. P-ValueModel 2 841.88 420.94Error 12 604.34 50.36Total 14 1446.23

8.36 0.001 P 0.01

Example: ANOVA (cont)7. This does give strong support (0.001 < P < 0.01) to

the claim that there is a difference in serotonin levels among the groups of men taking 0, 20, and 40 mg of Paxil. This does give strong support (0.001 < P < 0.01) to the claim that Paxil intake affects serotonin levels in young men.

Example: ANOVA Effect plot

0 5 10 15 20 25 30 35 40 4550

60

70

80

Amount of Paxil

Mea

n Se

roto

nin

leve

l

Problem with Multiple t tests

Overall Risk of Type I Error in Using Repeated t Tests at = 0.05

Table IX: Tukey = 0.05

Table IX: Tukey = 0.01

Example: TukeyA biologist wished to study the effects of ethanol on

sleep time. A sample of 20 rants matched for age and other characteristics, was selected, and each rat was given an oral injection having a particular concentration of ethanol per body weight. The rapid eye movement (REM) sleep time for each rat was then recorded for a 24-hour period with the results in the table on the next slide.

Does the data indicate that the true average REM sleep time depends of the concentration of ethanol?

Example: Tukey (cont)Treatment (conc. of ethanol) xNi si

0 (control) 88.6 73.2 91.4 68.0 75.2 79.28 10.181 g/kg 63.0 53.9 69.2 50.1 71.5 61.54 9.342 g/kg 44.9 59.5 40.2 56.3 38.7 47.92 9.464 g/kg 31.0 39.6 45.3 25.2 22.7 32.76 9.56

Source df SS MS F-Ratio P-ValueModel 3 5882.36 1960.79 21.09 0.0001Error 16 1487.40 92.96Total 19 7369.76

Example: Tukey (cont)i - j xNi - xNj Difference?

0 – 1 17.74 yes0 – 2 31.36 yes0 – 4 46.52 yes1 – 2 13.621 – 4 28.78 yes2 – 4 15.16

xN4 xN2 xN1 xN032.76 47.92 61.54 79.28

OUTPUT: Tukey cldiff The GLM Procedure Tukey's Studentized Range (HSD) Test for sleepNOTE: This test controls the Type I experimentwise error rate.

Alpha 0.05 Error Degrees of Freedom 16 Error Mean Square 92.9625 Critical Value of Studentized Range 4.04609 Minimum Significant Difference 17.446

Comparisons significant at the 0.05 level are indicated by ***.

Difference amount Between Simultaneous 95% Comparison Means Confidence Limits

1 - 2 17.740 0.294 35.186 *** 1 - 3 31.360 13.914 48.806 *** 1 - 4 46.520 29.074 63.966 *** 2 - 1 -17.740 -35.186 -0.294 *** 2 - 3 13.620 -3.826 31.066 2 - 4 28.780 11.334 46.226 *** 3 - 1 -31.360 -48.806 -13.914 *** 3 - 2 -13.620 -31.066 3.826 3 - 4 15.160 -2.286 32.606 4 - 1 -46.520 -63.966 -29.074 *** 4 - 2 -28.780 -46.226 -11.334 *** 4 - 3 -15.160 -32.606 2.286

OUTPUT: Tukey lines

The GLM Procedure Tukey's Studentized Range (HSD) Test for sleep NOTE: This test controls the Type I experimentwise error rate,

but it generally has a higher Type II error rate than REGWQ.

Alpha 0.05 Error Degrees of Freedom 16 Error Mean Square 92.9625 Critical Value of Studentized Range 4.04609 Minimum Significant Difference 17.446

Means with the same letter are not significantly different.

Tukey Grouping Mean N amount A 79.280 5 1

B 61.540 5 2 B C B 47.920 5 3 C C 32.760 5 4

Critical values for Dunnett’s Method

Example: DunnettA biologist wished to study the effects of ethanol on

sleep time. A sample of 20 rants matched for age and other characteristics, was selected, and each rat was given an oral injection having a particular concentration of ethanol per body weight. The rapid eye movement (REM) sleep time for each rat was then recorded for a 24-hour period with the results in the table on the next slide.

Does the data indicate that the true average REM sleep time is different for the ingestion of any alcohol?

Example: Dunnett (cont)Treatment (conc. of ethanol) xNi si

0 (control) 88.6 73.2 91.4 68.0 75.2 79.28 10.181 g/kg 63.0 53.9 69.2 50.1 71.5 61.54 9.342 g/kg 44.9 59.5 40.2 56.3 38.7 47.92 9.464 g/kg 31.0 39.6 45.3 25.2 22.7 32.76 9.56

Source df SS MS F-Ratio P-ValueModel 3 5882.36 1960.79 21.09 0.0001Error 16 1487.40 92.96Total 19 7369.76

Example: Dunnett (cont)

i - j xNi - xNj Difference?0 – 1 17.74 yes0 – 2 31.36 yes0 – 4 46.52 yes

OUTPUT: Dunnett cldiff The GLM Procedure Dunnett's t Tests for sleep NOTE: This test controls the Type I experimentwise error for comparisons of

all treatments against a control.

Alpha 0.05 Error Degrees of Freedom 16 Error Mean Square 92.9625 Critical Value of Dunnett's t 2.59232 Minimum Significant Difference 15.808

Comparisons significant at the 0.05 level are indicated by ***.

Difference amount Between Simultaneous 95% Comparison Means Confidence Limits

2 - 1 -17.740 -33.548 -1.932 *** 3 - 1 -31.360 -47.168 -15.552 *** 4 - 1 -46.520 -62.328 -30.712 ***

Randomized Block Experiments

Researchers are interested in the effect that acid has on growth rate of alfalfa plants. To control sunlight, the randomized block procedure is used.

ANOVA Table: RCBSource df SS MS F

Factor A(treatment) a – 1

Factor B(block) b - 1

Error (Within) (a-1)(b-1)

SST – SSA - SSB

Total n –1 = ab– 1

a2

ii 1

b (A x)

a b2

iji 1 j 1

(x x)

SSA

dfa

SSE

dfe

b2

jj 1

a (B x)

SSB

dfb

MSA

MSE

MSB

MSE

Stain 1 Stain 2 Stain 3Detergent 1 45 43 51Detergent 2 47 46 52Detergent 3 48 50 55Detergent 4 42 37 49

Example: RBDAn experiment was designed to study the performance

of four different detergents in cleaning clothes. The clothes were measured on cleanness (higher= cleaner) after being stained with three different common stains and cleaned with the four detergents. We want to know which if any detergent is better.

Example: RBD (cont)

1.Let 1 be the mean cleaning value for detergent 1.Let 2 be the mean cleaning value for detergent 2.Let 3 be the mean cleaning value for detergent 3.Let 4 be the mean cleaning value for detergent .

Example: RBD (cont)

2.H0: 1 = 2 = 3 = 4; mean cleaning levels for the 4 detergents are the same.HA: At least two of the detergents have different cleaning levels.

Example: RBD (cont)

7. This data does give strong support (P = 0.0063) to the claim that there is difference in how well the four detergents clean stains.

Source df SS MS F P ValueFactor A (soap) 3 110.92 36.97

Factor B (stain) 2 135.17 67.58

Error 6 18.83 3.14

Total 11 264.92

0.006311.78

Stain 1 Stain 2 Stain 3Detergent 1 45 43 51Detergent 2 47 46 52Detergent 3 48 50 55Detergent 4 42 37 49

Example: RBD 2An experiment was designed to study the performance

of four different detergents in cleaning clothes. The clothes were measured on cleanness (higher= cleaner) after being stained with three different common stains and cleaned with the four detergents. We want to know if the stains (blocks) have an effect.

Example: RBD 2 (cont)

1.Let 1 be the mean cleaning value for stain 1.

Let 2 be the mean cleaning value for stain 2.

Let 3 be the mean cleaning value for stain 3.

Example: RBD 2 (cont)

2.H0: 1 = 2 = 3 = 4; mean cleaning levels for the 4 detergents are the same.HA: At least two of the detergents have different cleaning levels.

Example: RBD 2 (cont)

7. This data does give strong support (P = 0.0018) to the claim that there is difference in how well the detergents clean the different stains.

Source df SS MS F P ValueFactor A (soap) 3 110.92 36.97

Factor B (stain) 2 135.17 67.58

Error 6 18.83 3.14

Total 11 264.92

0.001821.52

Example: RBD (cont)i - j xNi - xNj Difference?

1 – 2 2.0001 – 3 4.6671 – 4 3.6672 – 3 -2.6672 – 4 5.667 yes3 – 4 8.333 yes

xN4 xN1 xN2 xN342.67 46.33 48.33 51.00