Applied Statistics and Data Modeling Part 3: Analysis of Variance - - - PowerPoint PPT Presentation

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Applied Statistics and Data Modeling Part 3: Analysis of Variance - Balanced block designs Luc Duchateau 1 Paul Janssen 2 1 Faculty of Veterinary Medicine Ghent University, Belgium 2 Center for Statistics Hasselt University, Belgium 2020 UGent

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Applied Statistics and Data Modeling

Part 3: Analysis of Variance - Balanced block designs Luc Duchateau1 Paul Janssen2

1Faculty of Veterinary Medicine

Ghent University, Belgium

2Center for Statistics

Hasselt University, Belgium

2020

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Balanced block designs Overview

Blocking principle Different block designs Anova for RCB and RGCB

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Balanced block designs The blocking principle

The blocking principle

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Balanced block designs The blocking principle

The blocking principle

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Balanced block designs The blocking principle

The blocking principle

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Balanced block designs The blocking principle

The blocking principle

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Balanced block designs The blocking principle

The blocking principle

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Balanced block designs The blocking principle

The blocking principle

Group experimental units in homogeneous blocks Reduce background variability Blocking factor is known source of variability that is not of primary interest Attribute treatment at random to experimental units within a block

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Balanced block designs Different block designs

Example: Rice plants

Effect of insertion of a gene into a rice plant Three different types

Wild type (W) Haploid type (H) Diploid type (D)

Evaluation of growth for 1000-kernel weight (g) Greenhouse can be split up in different plots and blocks Temperature gradient from the door of the greenhouse at the left towards the right side of the greenhouse

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Balanced block designs Different block designs

Randomised design

Temperature

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Balanced block designs Different block designs

Randomised design

Temperature

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Balanced block designs Different block designs

Randomised design

Temperature

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Balanced block designs Different block designs

T emperature

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Balanced block designs Different block designs

T emperature

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Balanced block designs Different block designs

Randomised complete block design

Temperature

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Balanced block designs Different block designs

T emperature

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Balanced block designs Different block designs

Temperature

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Balanced block designs Different block designs

Randomised complete block design with repeated measures

Temperature

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Balanced block designs Different block designs

Temperature Temperature

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Balanced block designs Different block designs

Randomised complete block design with repeated measures Randomised generalised complete block design

Temperature Temperature

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Balanced block designs Different block designs

Randomised complete block design with repeated measures Randomised generalised complete block design Randomised complete block design

Temperature Temperature Temperature

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Balanced block designs Different block designs

Randomised complete block design with repeated measures Randomised generalised complete block design Randomised complete block design

Temperature Temperature Temperature

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Balanced block designs Different block designs

Temperature

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Balanced block designs Different block designs

Balanced incomplete block design

Temperature

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Balanced block designs Different block designs

T emperature

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Balanced block designs Different block designs

Balanced block design with repetition of W

T emperature

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Balanced block designs Different block designs

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Balanced block designs Different block designs

Balanced block design with different block sizes

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Balanced block designs Different block designs

Summary

RCB and GCB restrictive designs

Each block needs to contain exactly the same number of experimental units Number of experimental units needs to be a multiple of number of treatments

Balanced designs

Whenever we can not deduce information for the treatment effect by comparing blocks This is ok when the proportion of occurence of each treatment is the same in each block

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Balanced block designs ANOVA for balanced block designs

ANOVA for balanced block designs

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Balanced block designs ANOVA for balanced block designs

Factor effects model for RCB

Assume no interaction between treatment and block Model Yij = µ.. + αi + βj + eij where µ.. the overall population mean (constant) αi main effect of level i of factor A (treatment), i = 1, . . . , a constants with restriction

a

  • i=1

αi = 0 βj effect of block j, j = 1, . . . , b constants with restriction

b

  • j=1

βj = 0 eij independent random error term ∼ N(0, σ2)

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Balanced block designs ANOVA for balanced block designs

Factor effects model for RCB

Decomposition of sum of squares SStot = SStrt + SSblock + SSerr with SStot =

a

  • i=1

b

  • j=1
  • Yij − ¯

Y.. 2 SStrt =

a

  • i=1

b ¯

  • Yi. − ¯

Y.. 2 SSblock =

b

  • j=1

a ¯ Y.j − ¯ Y.. 2 SSerr =

a

  • i=1

b

  • j=1
  • Yij − ¯
  • Yi. − ¯

Y.j + ¯ Y.. 2

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Balanced block designs ANOVA for balanced block designs

Factor effects model for RCB

Mean sum of squares MStot = SStot ab − 1 MStrt = SStrt a − 1 MSblock = SSblock b − 1 MSerr = SSerr (a − 1)(b − 1)

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Balanced block designs ANOVA for balanced block designs

Factor effects model for RCB

Expected values E(MStrt) = σ2 + 1 a − 1

a

  • i=1

bα2

i

E(MSblock) = σ2 + 1 b − 1

b

  • j=1

aβ2

j

E(MSerr) = σ2 test statistic to test H0 : α1 = α2 = . . . = αa = 0 Ftrt* = MStrt MSerr ∼ Fa−1,(a−1)(b−1) P-value P(Ftrt* ≥ ftrt*)

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Balanced block designs ANOVA for balanced block designs

Example RCB

Block Plant type 1000-kernel weight (g) 1 W 22.4 1 H 24.8 1 D 25.2 2 W 27.3 2 H 28.6 2 D 28.4 3 W 24.5 3 H 25.8 3 D 26.2 4 W 23.3 4 H 23.5 4 D 24.1

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Balanced block designs ANOVA for balanced block designs

Example RCB

setwd("c:/users/lduchate/docs/OC/onderwijs/ADEKUS2020/") rcb <- read.delim("RicePlants_RCB.txt") str(rcb) ## 'data.frame': 12 obs. of 3 variables: ## $ Block : int 1 1 1 2 2 2 3 3 3 4 ... ## $ Type : Factor w/ 3 levels "D","H","W": 3 2 1 3 2 1 3 2 1 3 ... ## $ Weight: num 22.4 24.8 25.2 27.3 28.6 28.4 24.5 25.8 26.2 23.3 rcb$Block <- factor(rcb$Block) rcb$Type <- relevel(rcb$Type, "W")

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Balanced block designs ANOVA for balanced block designs

Example RCB

anova(lm(Weight ~ Block + Type, rcb)) ## Analysis of Variance Table ## ## Response: Weight ## Df Sum Sq Mean Sq F value Pr(>F) ## Block 3 36.036 12.0119 42.230 0.0001988 *** ## Type 2 5.787 2.8933 10.172 0.0118147 * ## Residuals 6 1.707 0.2844 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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Balanced block designs ANOVA for balanced block designs

Example RCB: What if we did not make use of blocks?

anova(lm(Weight ~ Type, rcb)) ## Analysis of Variance Table ## ## Response: Weight ## Df Sum Sq Mean Sq F value Pr(>F) ## Type 2 5.787 2.8933 0.6899 0.5263 ## Residuals 9 37.742 4.1936

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Balanced block designs ANOVA for balanced block designs

Example RCB - ANOVA table: What if we did not make use of blocks?

Term SS df MS f ∗ P(F≥ f ∗) Plant type 5.787 2 2.893 0.69 0.5263 Error 37.743 9 4.194 Total 43.529 11 3.957 Term SS df MS f ∗ P(F≥ f ∗) Plant type 5.787 2 2.893 10.17 0.0118 Block 36.036 3 12.012 42.23 0.0002 Error 1.707 6 0.284 Total 43.529 11 3.957

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Balanced block designs ANOVA for balanced block designs

Factor effects model for GCB

Block-treatment interaction can be added to the model Model Yijk = µ.. + αi + βj + (αβ)ij + eijk where

µ.. the overall population mean (constant) αi main effect of level i of factor A treatment, i = 1, . . . , a constants with restriction

a

  • i=1

αi = 0 βj effect of block j, j = 1, . . . , b constants with restriction

b

  • j=1

βj = 0 (αβ)ij interaction between level i of factor A and block j constants with restrictions

a

  • i=1

(αβ)ij = 0

b

  • j=1

(αβ)ij = 0 eijk independent random error term ∼ N(0, σ2), k = 1, . . . , n

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Balanced block designs ANOVA for balanced block designs

Factor effects model for GCB

Decomposition of sum of squares SStot = SStrt + SSblock + SStrt*block + SSerr with

SStot =

a

  • i=1

b

  • j=1

n

  • k=1
  • Yijk − ¯

Y... 2 SStrt = nb

a

  • i=1

¯ Yi.. − ¯ Y... 2 SSblock = na

b

  • j=1

¯ Y.j. − ¯ Y... 2 SStrt*block = n

a

  • i=1

b

  • j=1

¯

  • Yij. − ¯

Yi.. − ¯ Y.j. + ¯ Y... 2 SSerr =

a

  • i=1

b

  • j=1

n

  • k=1
  • Yijk − ¯

Yij. 2

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Balanced block designs ANOVA for balanced block designs

Factor effects model for GCB

Mean sum of squares MStot = SStot nab − 1 MStrt = SStrt a − 1 MSblock = SSblock b − 1 MStrt*block = SStrt*block (a − 1)(b − 1) MSerr = SSerr (n − 1)ab

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Balanced block designs ANOVA for balanced block designs

Factor effects model for GCB

Expected values

E(MStrt) = σ2 + nb

a

  • i=1

α2

i

a − 1 E(MSblock) = σ2 + na

b

  • j=1

β2

j

b − 1 E(MStrt*block) = σ2 + n

a

  • i=1

b

  • j=1

(αβ)2

ij

(a − 1)(b − 1) E(MSerr) = σ2

Test statistic to test H0 : α1 = α2 = . . . = αa = 0

Ftrt = MStrt MSerr ∼ Fa−1,(n−1)ab

P-value:

P(Ftrt ≥ ftrt)

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Balanced block designs ANOVA for balanced block designs

Example GCB

1000-kernel weight (g) Block Plant type Observation 1 Observation 2 1 W 22.4 23.0 1 H 24.8 25.2 1 D 25.2 25.6 2 W 27.3 26.3 2 H 28.6 27.5 2 D 28.4 29.3 3 W 24.5 24.8 3 H 25.8 25.9 3 D 26.2 25.3 4 W 23.3 23.9 4 H 23.5 23.9 4 D 24.1 26.2

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Balanced block designs ANOVA for balanced block designs

Example GCB

gcb <- read.delim("RicePlants_GCB.txt") str(gcb) ## 'data.frame': 24 obs. of 3 variables: ## $ Block : int 1 1 1 2 2 2 3 3 3 4 ... ## $ Type : Factor w/ 3 levels "D","H","W": 3 2 1 3 2 1 3 2 1 3 ... ## $ Weight: num 22.4 24.8 25.2 27.3 28.6 28.4 24.5 25.8 26.2 23.3 gcb$Block <- factor(gcb$Block) gcb$Type <- relevel(gcb$Type, "W")

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Balanced block designs ANOVA for balanced block designs

Example GCB

anova(lm(Weight ~ Block * Type, gcb)) ## Analysis of Variance Table ## ## Response: Weight ## Df Sum Sq Mean Sq F value Pr(>F) ## Block 3 53.202 17.7339 44.6136 8.774e-07 *** ## Type 2 14.131 7.0654 17.7746 0.0002584 *** ## Block:Type 6 3.416 0.5693 1.4322 0.2803302 ## Residuals 12 4.770 0.3975 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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Balanced block designs ANOVA for balanced block designs

Example GCB: What if we did not make use of blocks?

anova(lm(Weight ~ Type, gcb)) ## Analysis of Variance Table ## ## Response: Weight ## Df Sum Sq Mean Sq F value Pr(>F) ## Type 2 14.131 7.0654 2.417 0.1136 ## Residuals 21 61.387 2.9232

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Balanced block designs ANOVA for balanced block designs

Example GCB - ANOVA table: What if we did not make use of blocks?

Term SS df MS f ∗ P(F≥ f ∗) Plant type 14.131 2 7.065 2.42 0.1136 Error 61.388 21 2.923 Total 75.518 23 3.283 Term SS df MS f ∗ P(F≥ f ∗) Plant type 14.131 2 7.065 17.77 0.0003 Block 53.202 3 17.734 44.61 <0.0001 Plant type*Block 3.416 6 0.569 1.43 0.280 Error 4.770 12 0.397 Total 75.518 23 3.283

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Problem 1.

In an animal welfare study researchers would like to investigate whether voluntary water intake could be a good indicator to detect thirst in chickens. They select four different places in the stable and house at each place three chickens in individual cages. In each place one chicken has continuously access to water, one chicken is deprived of water during 3 hours and the third chicken is deprived of water during 6 hours. After their treatment (continuous access to water, deprivation for 3 hours, deprivation for 6 hours) the water intake of the chickens during 30 minutes is measured. Which of the following statements is true? The researchers did not use blocks in their experiment. Place in the stable and duration of deprivation are both factors of interest In this experiment the cages are the blocks and the chicken inside the cage is the experimental unit. In this experiment the place in the stable is the block and the researchers are interested in the effect of duration of deprivation on the voluntary water intake In this experiment, there are four blocks and three treatment groups.

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Problem 2.

Researchers want to test three different types of eye drops (A,B,C) to treat conjunctivitis in cats. 10 cats suffering from conjunctivitis in both eyes are included in the experiment. They designed the experiment as follows: 1 2 3 4 5 6 7 8 9 10 Left eye A A C B B C A C C A Right eye B C B C A A B B A B What type of design did the researchers use? Randomised complete block design Balanced incomplete block design Balanced block design with different block sizes None of the above

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Problem 3.

Consider again the previous experiment: researchers want to test three different types of eye drops (A,B,C) to treat conjunctivitis in cats. 10 cats suffering from conjunctivitis in both eyes are included in the experiment. They designed the experiment as follows: 1 2 3 4 5 6 7 8 9 10 Left eye A A C B B C A C C A Right eye B C B C A A B B A B Which of the following statements is correct? If the researchers would have used only 9 cats, removing cat 10 from the experiment, the design of the experiment would be a balanced incomplete block design. This experiment includes 10 experimental units. This experiment is definitely not randomized since treatment C is assigned more

  • ften to the left eye than the right eye.

By comparing cat 5 and 6 we can deduce information about the treatment effect

  • f treatment A.

None of the above

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Problem 4.

Which of the following designs is balanced? Double lines depict blocks.

  • A

D B C C A D A C B B D

  • A

C B B C A C B B B B B C A B A A A A B B C B C B B B C

  • A

B A A C C B B B C A C None of the above

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Problem 5.

In an animal welfare study researchers would like to investigate whether voluntary water intake could be a good indicator to detect thirst in chickens. They select four different places in the stable and house at each place three chickens in individual cages. In each place one chicken has continuously access to water, one chicken is deprived of water during 3 hours and the third chicken is deprived of water during 6 hours. After their treatment (continuous access to water, deprivation for 3 hours, deprivation for 6 hours) the water intake of the chickens during 30 minutes is measured. The P-value for testing the null hypothesis that there is no difference in water intake between the three deprivation groups when taking into account the place in the stable is equal to 0.00188, the total sum of squares is 171.35 and the error sum of squares is 17.96.

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What would be the P-value for testing that same null hypothesis when the researchers ignored the blocks in the analysis of their experiment? P=0.0079 P=0.0169 P=0.002 There is not enough information to calculate the P-value.

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Data for the water deprivation experiment can be found in ”thirst2.csv”. Considering all pairwise two-sided comparisons between the treatment groups with a Bonferroni correction, which of the following statements are true? The effect of deprivation on the water intake is significant (p=0.000807). The effect of deprivation on the water intake is significant (p=0.0000204). There is a significant difference between the control group and the group with 3 hours of deprivation (p=0.004) and between the control group and the group with 6 hours of deprivation (p=0.0000158). There is a significant difference between the control group and the group with 6 hours of deprivation (p=0.000605) .

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If we are specifically interested in investigating whether the deprivation groups drink more water than the control group, which of the following statements are correct (use Bonferroni correction)? Chickens in both deprivation groups drink significantly more water (3h: p=0.00405, 6h: p= 0.0000158) than chickens in the control group. Chickens in both deprivation groups drink significantly more water (3h: p=0.00135, 6h: p= 0.00000527) than chickens in the control group. Chickens in both deprivation groups drink significantly more water (3h: p=0.0273, 6h: p= 0.000202) than chickens in the control group. Only the chickens in the 6h deprivation group drink significantly more water (p=0.000605) than the control group.

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