Topic 24: Two-Way ANOVA
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Transcript of Topic 24: Two-Way ANOVA
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Topic 24: Two-Way ANOVA
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Outline
• Two-way ANOVA
–Data
–Cell means model
–Parameter estimates
–Factor effects model
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Two-Way ANOVA
• The response variable Y is continuous
• There are two categorical explanatory variables or factors
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Data for two-way ANOVA
• Y is the response variable
• Factor A with levels i = 1 to a
• Factor B with levels j = 1 to b
• Yijk is the kth observation in cell (i,j)
• In Chapter 19, we assume equal
sample size in each cell (nij=n)
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KNNL Example• KNNL p 833
• Y is the number of cases of bread sold
• A is the height of the shelf display, a=3 levels: bottom, middle, top
• B is the width of the shelf display, b=2 levels: regular, wide
• n=2 stores for each of the 3x2=6 treatment combinations (nT=12)
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Read the data
data a1; infile ‘../data/ch19ta07.txt'; input sales height width;
proc print data=a1; run;
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The dataObs sales height width 1 47 1 1 2 43 1 1 3 46 1 2 4 40 1 2 5 62 2 1 6 68 2 1 7 67 2 2 8 71 2 2 9 41 3 1 10 39 3 1 11 42 3 2 12 46 3 2
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Notation
• For Yijk we use
– i to denote the level of the factor A
– j to denote the level of the factor B
–k to denote the kth observation in cell (i,j)
• i = 1, . . . , a levels of factor A
• j = 1, . . . , b levels of factor B
• k = 1, . . . , n observations in cell (i,j)
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Model
• We assume that the response variable observations are
–Normally distributed
•With a mean that may depend on the levels of the factors A and B
•With a constant variance
– Independent
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Cell Means Model• Yijk = μij + εijk
–where μij is the theoretical mean or expected value of all observations in cell (i,j) – the εijk are iid N(0, σ2)
• This means Yijk ~ N(μij, σ2), independent• The parameters of the model are– μij, for i = 1 to a and j = 1 to b –σ2
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Estimates• Estimate μij by the mean of the
observations in cell (i,j), • • For each (i,j) combination, we can get
an estimate of the variance
• • We need to combine these to get an
estimate of σ2
ij.Yn/)Y(Y k ijkij.
k2
ij.ijk2ij )1n/()YY(s
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Pooled estimate of σ2
• In general we pool the sij2, using
weights proportional to the df, nij -1
• The pooled estimate is
s2 = (Σ (nij-1)sij2) / (Σ(nij-1))
• Here, nij = n, so s2 = (Σsij2) / (ab),
which is the average sample variance
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Run proc glm
proc glm data=a1; class height width; model sales= height width height*width; means height width height*width;run;
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Output
Class Level InformationClass Levels Valuesheight 3 1 2 3width 2 1 2
Number of Observations Read 12Number of Observations Used 12
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Means statement height
Level ofheight N
sales
Mean Std Dev1 4 44.0000000 3.16227766
2 4 67.0000000 3.74165739
3 4 42.0000000 2.94392029
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Means statement width
Level ofwidth N
sales
Mean Std Dev1 6 50.0000000 12.0664825
2 6 52.0000000 13.4313067
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Means statement ht*w
Level ofheight
Level ofwidth N
sales
Mean Std Dev1 1 2 45.0000000 2.828427121 2 2 43.0000000 4.242640692 1 2 65.0000000 4.242640692 2 2 69.0000000 2.828427123 1 2 40.0000000 1.414213563 2 2 44.0000000 2.82842712
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Code the factor levelsdata a1; set a1; if height eq 1 and width eq 1 then hw='1_BR'; if height eq 1 and width eq 2 then hw='2_BW'; if height eq 2 and width eq 1 then hw='3_MR'; if height eq 2 and width eq 2 then hw='4_MW'; if height eq 3 and width eq 1 then hw='5_TR'; if height eq 3 and width eq 2 then hw='6_TW';
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Plot the data
symbol1 v=circle i=none;proc gplot data=a1; plot sales*hw/frame;run;
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The plot
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Put the means in a2
proc means data=a1; var sales; by height width; output out=a2 mean=avsales;proc print data=a2; run;
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Output Data Set
Obs height width _TYPE_ _FREQ_ avsales
1 1 1 0 2 45 2 1 2 0 2 43 3 2 1 0 2 65 4 2 2 0 2 69 5 3 1 0 2 40 6 3 2 0 2 44
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Plot the means
symbol1 v=square i=join c=black;symbol2 v=diamond i=join c=black;proc gplot data=a2; plot avsales*height=width/frame;run;
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The interaction plot
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Questions to consider
• Does the height of the display affect
sales? If yes, compare top with middle,
top with bottom, and middle with bottom
• Does the width of the display affect
sales? If yes, compare regular and wide
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But wait!!! Are these factor level comparisons
meaningful?• Does the effect of height on sales
depend on the width?
• Does the effect of width on sales depend on the height?
• If yes, we have an interaction and we need to do some additional analysis
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Factor effects model
• For the one-way ANOVA model, we wrote μi = μ + αi
• Here we use μij = μ + αi + βj + (αβ)ij
• Under “common” formulation– μ (μ.. in KNNL) is the “overall mean”
– αi is the main effect of A
– βj is the main effect of B
– (αβ)ij is the interaction between A and B
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Factor effects model
• μ = (Σij μij)/(ab)
• μi. = (Σj μij)/b and μ.j = (Σi μij)/a
• αi = μi. – μ and βj = μ.j - μ
• (αβ)ij is difference between μij and μ + αi + βj
• (αβ)ij = μij - (μ + (μi. - μ) + (μ.j - μ))
= μij – μi. – μ.j + μ
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Interpretation
• μij = μ + αi + βj + (αβ)ij
• μ is the “overall” mean
• αi is an adjustment for level i of A
• βj is an adjustment for level j of B
• (αβ)ij is an additional adjustment that takes into account both i and j that cannot be explained by the previous adjustments
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Constraints for this framework
• α. = Σi αi= 0
• β. = Σjβj = 0
• (αβ).j = Σi (αβ)ij = 0 for all j
• (αβ)i. = Σj (αβ)ij = 0 for all i
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Estimates for factor effects model
....j...i.ijij
....j.j.....ii
.j..j..i.i
ijk ijk...
YYYY)ˆ(
YYˆ and YYˆ
Yˆ and Yˆ
abn/)Y(Yˆ
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SS for ANOVA Table22
.. ...ijk
2jijk
2ijijk
2ijk .ijk
2...ijk
ˆSSA (Y Y )
ˆSSB
SSAB ( )
SSE (Y Y )
SSTO (Y Y )
ˆ
i iijk
ij
ijk
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df for ANOVA Table
• dfA = a-1
• dfB = b-1
• dfAB = (a-1)(b-1)
• dfE = ab(n-1)
• dfT = abn-1 = nT-1
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MS for ANOVA Table
• MSA = SSA/dfA
• MSB = SSB/dfB
• MSAB = SSAB/dfAB
• MSE = SSE/dfE
• MST = SST/dfT
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Hypotheses for two-way ANOVA
• H0A: αi = 0 for all i
• H1A: αi ≠ 0 for at least one i
• H0B: βj = 0 for all j
• H1B: βj ≠ 0 for at least one j
• H0AB: (αβ)ij = 0 for all (i,j)
• H1AB: (αβ)ij ≠ 0 for at least one (i,j)
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F statistics
• H0A is tested by FA = MSA/MSE; df=dfA, dfE
• H0B is tested by FB = MSB/MSE; df=dfB,
dfE
• H0AB is tested by FAB = MSAB/MSE;
df=dfAB, dfE
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ANOVA Table
Source df SS MS F A a-1 SSA MSA MSA/MSE B b-1 SSB MSB MSB/MSE AB (a-1)(b-1) SSAB MSAB MSAB/MSEError ab(n-1) SSE MSE _ Total abn-1 SSTO MST
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P-values
• P-values are calculated using the F(dfNumerator, dfDenominator) distributions
• If P ≤ 0.05 we conclude that the effect being tested is statistically significant
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KNNL Example• NKNW p 833• Y is the number of cases of bread sold• A is the height of the shelf display, a=3
levels: bottom, middle, top• B is the width of the shelf display, b=2:
regular, wide• n=2 stores for each of the 3x2
treatment combinations
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PROC GLM
proc glm data=a1; class height width; model sales= height width height*width;run;
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Output
Note that there are 6 cells inthis design…(6-1)df for model
Source DFSum of
SquaresMean
Square F Value Pr > FModel 5 1580.0000 316.000000 30.58 0.0003Error 6 62.000000 10.333333Corrected Total
11 1642.0000
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Output ANOVA
Note Type I and Type III Analyses are the same becausenij is constant
Source DF Type III SS Mean Square F Value Pr > Fheight 2 1544.00000 772.000000 74.71 <.0001
width 1 12.000000 12.000000 1.16 0.3226
height*width 2 24.000000 12.000000 1.16 0.3747
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Other output
R-Square Coeff Var Root MSE sales Mean0.962241 6.303040 3.214550 51.00000
Commonly do not consider R-sq when performing ANOVA…interested more in difference in levels rather than the models predictive ability
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Results
• The main effect of height is statistically significant (F=74.71; df=2,6; P<0.0001)
• The main effect of width is not statistically significant (F=1.16; df=1,6; P=0.32)
• The interaction between height and width is not statistically significant (F=1.16; df=2,6; P=0.37)
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Interpretation
• The height of the display affects sales of bread
• The width of the display has no apparent effect
• The effect of the height of the display is similar for both the regular and the wide widths
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Plot of the means
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Additional analyses
• We will need to do additional analyses to explain the height effect (factor A)
• There were three levels: bottom, middle and top
• We could rerun the data with a one-way anova and use the methods we learned in the previous chapters
• Use means statement with lines
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Run Proc GLM
proc glm data=a1; class height width; model sales= height width height*width; means height / tukey lines; lsmeans height / adjust=tukey;run;
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MEANS OutputAlpha 0.05Error Degrees of Freedom 6Error Mean Square 10.33333Critical Value of Studentized Range 4.33920Minimum Significant Difference 6.9743
Means with the same letter are not significantly different.
Tukey Grouping Mean N heightA 67.000 4 2
B 44.000 4 1BB 42.000 4 3
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LSMEANS Outputheight sales LSMEAN
LSMEAN Number
1 44.0000000 1
2 67.0000000 2
3 42.0000000 3
Least Squares Means for effect heightPr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: salesi/j 1 2 31 0.0001 0.67142 0.0001 <.00013 0.6714 <.0001
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Last slide
• We went over Chapter 19
• We used program topic24.sas to generate the output for today.