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Analysis of variance NORBERTO E. MILLA Department of Statistics Visayas State University [email protected]
Outline
1. Review of basic features of CRD experiments
2. What is analysis of variance?
3. ANOVA for CRD experiments
4. Assumptions of ANOVA
5. Post hoc analysis
Completely Randomized Design
Frequently used to compare treatments when environmental conditions are fairly uniform.
Each treatment is applied at random to several experimental units
Response=mean + treatment effect + error
, 1,2,..., ; 1,2,...,ij i ij iY i t j r
),0(~ 2
NIDij
Completely Randomized Design
Consider an experiment with 6 feed treatments (T1 to T6). Each treatment is applied to 3 pens with each pen having 5 chicks.
T2
T1
T5
T6T3
T4T5
T5 T1 T4
T6
T3T2T6
T1
T3
T4
T2
What is analysis of variance?
A method of partitioning the total variance in the response into different components which can be attributed to different sources
Systematic variation-effect of manipulated factors; and random variation (experimental error)
Method of comparing the effect of treatments on the response variable
Method of comparing the means of three or more treatments
What is analysis of variance?
Pesticide Yield
None 55
None 45
None 46
Biological 64
Biological 52
Biological 42
Chemical 65
Chemical 52
Chemical 66
2222 1.54661.5445)1.5455( Ts
Pesticide Mean
None 48.7
Biological 52.7
Chemical 61.0
Grand Mean 54.1
2222 1.540.611.547.52)1.547.48( Bs
2222 7.48467.4845)7.4855( Ws
222 7.52427.5252)7.5264(
222 0.61660.6152)0.6165(
222
WBT sss
systematic random
What is analysis of variance?
Compare systematic variation with random variation (experimental error)
In CRD, variation between treatments (MSTr) is compared with random variation (MSE)
Test statistic: F
𝑭 =𝑴𝑺𝑻𝒓
𝑴𝑺𝑬
p-value- the probability of finding the observed (ormore extreme) sample results (test statistic) , assumingthe null hypothesis is true
small p-value suggests strong the evidence thatyou should reject the null hypothesis
Guide:
p>0.05do not reject Ho”non-significant effect”
0.05≤p<0.01reject Ho”significant at 5% level”
p≤0.01reject Ho ”significant at 1% level”
“highly significant”
The p-value approach
Completely Randomized Design
An experiment was conducted to test the effects of addition ofvarious sugars on the growth of pea plants. Pea plants wererandomly assigned to 5 treatment groups: Control (no sugaradded), 2% glucose, 2% fructose, 1% glucose+1%fructose,and 2% sucrose. The data collected is length of pea plants inocular units (x0.114 mm). The data is provided in the nextslide.
Completely Randomized Design
Completely Randomized Design
ANOVA for CRD
Menu: Statistics>Linear models and related>ANOVA/MANOVA
>One-way ANOVA
ANOVA for CRD
Menu: Statistics>Linear models and related>ANOVA/MANOVA
>One-way ANOVA
Bartlett's test for equal variances: chi2(4) = 13.9386 Prob>chi2 = 0.007
Total 1322.82 49 26.9963265
Within groups 245.5 45 5.45555556
Between groups 1077.32 4 269.33 49.37 0.0000
Source SS df MS F Prob > F
Analysis of Variance
. oneway length trtcode
ANOVA for CRD
Menu: Statistics>Linear models and related>ANOVA/MANOVA
>Analysis of variance and covariance
ANOVA for CRD
Menu: Statistics>Linear models and related>ANOVA/MANOVA
>Analysis of variance and covariance
Total 1322.82 49 26.996327
Residual 245.5 45 5.4555556
trtcode 1077.32 4 269.33 49.37 0.0000
Model 1077.32 4 269.33 49.37 0.0000
Source Partial SS df MS F Prob>F
Root MSE = 2.33571 Adj R-squared = 0.7979
Number of obs = 50 R-squared = 0.8144
Recall:
is estimated by the residuals ( )
Normality of the residuals
Homogeneity of the variances of the residuals
Independence of observations
Linear relationship between response and theindependent variable(s)
Assumptions of the ANOVA
),0(~ 2
NIDij
ij ijijij YYe
Normality of the residuals
Tests: Shapiro-Wilk, Shapiro-Francia
Homogeneity of the variances of the residuals
Tests: Bartlett’s, Levene’s, Brown-and-Forsythe’s
Independence of observations
Proper randomization ensures independence
Linear relationship between response and theindependent variable(s)
Graphical approaches can help you decide onfunctional form of the model
Assumptions of the ANOVA
MENU: Statistics>Postestimation
Generating residuals
MENU: Statistics>Summaries, tables and tests>distributional plots and tests>Shapiro-Wilk normality test
Testing normality of residuals
MENU: Statistics>Summaries, tables and tests>distributional plots and tests>Shapiro-Wilk normality test
Testing normality of residuals
residuals 10 0.90476 1.468 0.684 0.24691
Variable Obs W V z Prob>z
Shapiro-Wilk W test for normal data
-> treatment = Control
residuals 10 0.95858 0.638 -0.737 0.76953
Variable Obs W V z Prob>z
Shapiro-Wilk W test for normal data
-> treatment = Fructose2%
MENU: Statistics>Summaries, tables and tests>Classical tests of hypothesis>Robust equal-variance test
Test of homogeneity of variance of residuals
MENU: Statistics>Summaries, tables and tests>Classical tests of hypothesis>Robust equal-variance test
Test of homogeneity of variance of residuals
W10 = 5.5602003 df(4, 45) Pr > F = 0.00100461
W50 = 4.5767359 df(4, 45) Pr > F = 0.00346787
W0 = 5.9102420 df(4, 45) Pr > F = 0.00065483
Total 5.960e-10 1.0101525 50
Sucrose2% 1.080e-08 .80869902 10
Glucose2% 1.192e-08 .73849323 10
Glucose1%_Fructos.. -2.980e-09 .63822563 10
Fructose2% -7.451e-10 .84563224 10
Control -1.602e-08 1.7982667 10
Treatment Mean Std. Dev. Freq.
Summary of Standardized residuals
Observing significant differences in the meanresponse among treatments requires furtheranalysis
Where do the differences lie? Is T1 significantlydifferent from T3? Is the control different from theexperimental treatments?
Multiple comparison procedures (post hoc)
Fisher’s Least Significant Difference (LSD)
Tukey’s Honest Significant Difference (HSD)
Duncan Multiple Range Test (DMRT)
Post hoc analysis
ANOVA for CRD
Total 1322.82 49 26.996327
Residual 245.5 45 5.4555556
trtcode 1077.32 4 269.33 49.37 0.0000
Model 1077.32 4 269.33 49.37 0.0000
Source Partial SS df MS F Prob>F
Root MSE = 2.33571 Adj R-squared = 0.7979
Number of obs = 50 R-squared = 0.8144
Sucrose2% 64.1
Glucose2% 59.3
Glucose1%_Fructo 58
Fructose2% 58.2
Control 70.1
treatment mean
MENU: Statistics>Postestimation
Post hoc analysis
Post hoc analysis
Post hoc analysis: Tukey
not significantly different at the 5% level.
Note: Margins sharing a letter in the group label are
Sucrose2% 64.1 .7386173
Glucose2% 59.3 .7386173 A
Glucose1%_Fructose1% 58 .7386173 A
Fructose2% 58.2 .7386173 A
Control 70.1 .7386173
trt
Margin Std. Err. Groups
Tukey
trt 10
Comparisons
Number of
Margins : asbalanced
Pairwise comparisons of marginal linear predictions
Type of Sugar Mean Length
Control 70.1a
2% Glucose 59.3c
2% Fructose 58.2c
1% Glucose + 1% Fructose 58.0c
2% Sucrose 64.1b
Post hoc analysis: Dunnet
Sucrose2% vs Control -6 1.044563 -5.74 0.000
Glucose2% vs Control -10.8 1.044563 -10.34 0.000
Glucose1%_Fructose1% vs Control -12.1 1.044563 -11.58 0.000
Fructose2% vs Control -11.9 1.044563 -11.39 0.000
trt
Contrast Std. Err. t P>|t|
Dunnett
