chapter 8 1-way analysis of variance - completely randomized design
TRANSCRIPT
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Chapter 8
1-Way Analysis of Variance - Completely Randomized Design
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Comparing t > 2 Groups - Numeric Responses
• Extension of Methods used to Compare 2 Groups• Independent Samples and Paired Data Designs• Normal and non-normal data distributions
DataDesign
Normal Non-normal
IndependentSamples(CRD)
F-Test1-WayANOVA
Kruskal-Wallis Test
Paired Data(RBD)
F-Test2-WayANOVA
Friedman’sTest
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Completely Randomized Design (CRD)
• Controlled Experiments - Subjects assigned at random to one of the t treatments to be compared
• Observational Studies - Subjects are sampled from t existing groups
• Statistical model yij is measurement from the jth subject from group i:
ijiijiijy
where is the overall mean, i is the effect of treatment i , ij is a random error, and i is the population mean for group i
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1-Way ANOVA for Normal Data (CRD)
• For each group obtain the mean, standard deviation, and sample size:
1
)( 2.
.
i
jiij
ii
jij
i n
yy
sn
y
y
• Obtain the overall mean and sample size
N
y
N
ynynynnN i j ijtt
t
..11
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......
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Analysis of Variance - Sums of Squares
• Total Variation
1)(1 1
2..
NdfyyTSS Total
k
i
n
j iji
• Between Group (Sample) Variation
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• Within Group (Sample) Variation
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tNdfsnyySSE i
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Analysis of Variance Table and F-TestSource ofVariation Sum of Squares
Degrres ofFreedom Mean Square F
Treatments SST t-1 MST=SST/(t-1) F=MST/MSEError SSE N-t MSE=SSE/(N-t)Total TSS N-1
• Assumption: All distributions normal with common variance
•H0: No differences among Group Means (t =0)
• HA: Group means are not all equal (Not all i are 0)
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TableFFRRMSE
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Expected Mean Squares
• Model: yij = +i + ij with ij ~ N(0,2), i = 0:
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Expected Mean Squares
• 3 Factors effect magnitude of F-statistic (for fixed t)– True group effects (1,…,t)
– Group sample sizes (n1,…,nt)
– Within group variance (2)
• Fobs = MST/MSE
• When H0 is true (1=…=t=0), E(MST)/E(MSE)=1 • Marginal Effects of each factor (all other factors fixed)
– As spread in (1,…,t) E(MST)/E(MSE)
– As (n1,…,nt) E(MST)/E(MSE) (when H0 false)
– As 2 E(MST)/E(MSE) (when H0 false)
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A)=100, 1=-20, 2=0, 3=20, = 20 B)=100, 1=-20, 2=0, 3=20, = 5
C)=100, 1=-5, 2=0, 3=5, = 20 D)=100, 1=-5, 2=0, 3=5, = 5
n A B C D4 9 129 1.5 98 17 257 2 1712 25 385 2.5 2520 41 641 3.5 41
)(
)(
MSEE
MSTE
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Example - Seasonal Diet Patterns in Ravens
• “Treatments” - t = 4 seasons of year (3 “replicates” each)– Winter: November, December, January
– Spring: February, March, April
– Summer: May, June, July
– Fall: August, September, October
• Response (Y) - Vegetation (percent of total pellet weight)• Transformation (For approximate normality):
100arcsin'
YY
Source: K.A. Engel and L.S. Young (1989). “Spatial and Temporal Patterns in the Diet of Common Ravens in Southwestern Idaho,” The Condor, 91:372-378
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Seasonal Diet Patterns in Ravens - Data/MeansY Winter(i=1) Fall(i=2) Summer(i=3) Fall (i=4)
j=1 94.3 80.7 80.5 67.8j=2 90.3 90.5 74.3 91.8j=3 83.0 91.8 32.4 89.3
Y' Winter(i=1) Fall(i=2) Summer(i=3) Fall (i=4)j=1 1.329721 1.115957 1.113428 0.967390j=2 1.254080 1.257474 1.039152 1.280374j=3 1.145808 1.280374 0.605545 1.237554
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Seasonal Diet Patterns in Ravens - Data/MeansPlot of Transformed Data by Season
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Seasonal Diet Patterns in Ravens - ANOVA
241038.0)161773.1237554.1(...)243203.1329721.1(
8)4-12(df :Variation GroupWithin
197387.0)135572.1161773.1(...)135572.124203.1(3
3)1-4(df :Variation GroupBetween
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11)1-12(df :Variation Total
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Total
SSE
SST
TSS
ANOVASource of Variation SS df MS F P-value F critBetween Groups 0.197387 3 0.065796 2.183752 0.167768 4.06618Within Groups 0.241038 8 0.03013
Total 0.438425 11
Do not conclude that seasons differ with respect to vegetation intake
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Seasonal Diet Patterns in Ravens - Spreadsheet
Month Season Y' Season MeanOverall Mean TSS SST SSENOV 1 1.329721 1.243203 1.135572 0.037694 0.011584 0.007485DEC 1 1.254080 1.243203 1.135572 0.014044 0.011584 0.000118JAN 1 1.145808 1.243203 1.135572 0.000105 0.011584 0.009486FEB 2 1.115957 1.217935 1.135572 0.000385 0.006784 0.010400MAR 2 1.257474 1.217935 1.135572 0.014860 0.006784 0.001563APR 2 1.280374 1.217935 1.135572 0.020968 0.006784 0.003899MAY 3 1.113428 0.919375 1.135572 0.000490 0.046741 0.037657JUN 3 1.039152 0.919375 1.135572 0.009297 0.046741 0.014346JUL 3 0.605545 0.919375 1.135572 0.280928 0.046741 0.098489AUG 4 0.967390 1.161773 1.135572 0.028285 0.000687 0.037785SEP 4 1.280374 1.161773 1.135572 0.020968 0.000687 0.014066OCT 4 1.237554 1.161773 1.135572 0.010400 0.000687 0.005743
Sum 0.438425 0.197387 0.241038
Total SS Between Season SS Within Season SS
(Y’-Overall Mean)2 (Group Mean-Overall Mean)2 (Y’-Group Mean)2
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CRD with Non-Normal Data Kruskal-Wallis Test
• Extension of Wilcoxon Rank-Sum Test to k > 2 Groups• Procedure:
– Rank the observations across groups from smallest (1) to largest ( N = n1+...+nk ), adjusting for ties
– Compute the rank sums for each group: T1,...,Tk . Note that T1+...+Tk = N(N+1)/2
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Kruskal-Wallis Test
• H0: The k population distributions are identical (1=...=k)
• HA: Not all k distributions are identical (Not all i are equal)
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An adjustment to H is suggested when there are many ties in the data. Formula is given on page 426 of O&L.
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Example - Seasonal Diet Patterns in RavensMonth Season Y' RankNOV 1 1.329721 12DEC 1 1.254080 8JAN 1 1.145808 6FEB 2 1.115957 5MAR 2 1.257474 9APR 2 1.280374 10.5MAY 3 1.113428 4JUN 3 1.039152 3JUL 3 0.605545 1AUG 4 0.967390 2SEP 4 1.280374 10.5OCT 4 1.237554 7
• T1 = 12+8+6 = 26
• T2 = 5+9+10.5 = 24.5
• T3 = 4+3+1 = 8
• T4 = 2+10.5+7 = 19.5
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Post-hoc Comparisons of Treatments
• If differences in group means are determined from the F-test, researchers want to compare pairs of groups. Three popular methods include:– Fisher’s LSD - Upon rejecting the null hypothesis of no
differences in group means, LSD method is equivalent to doing pairwise comparisons among all pairs of groups as in Chapter 6.
– Tukey’s Method - Specifically compares all t(t-1)/2 pairs of groups. Utilizes a special table (Table 10, pp. 1110-1111).
– Bonferroni’s Method - Adjusts individual comparison error rates so that all conclusions will be correct at desired confidence/significance level. Any number of comparisons can be made. Very general approach can be applied to any inferential problem
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Fisher’s Least Significant Difference Procedure
• Protected Version is to only apply method after significant result in overall F-test
• For each pair of groups, compute the least significant difference (LSD) that the sample means need to differ by to conclude the population means are not equal
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Tukey’s W Procedure
• More conservative than Fisher’s LSD (minimum significant difference and confidence interval width are higher).
• Derived so that the probability that at least one false difference is detected is (experimentwise error rate)
. .
. .
( , ) given in Table 10, pp. 1110-1111 with
Conclude if
Tukey's Confidence Interval:
( , ) 1 1When the sample sizes are unequal, use
2
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y y W
y y W
q tW MSE
n n
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Bonferroni’s Method (Most General)• Wish to make C comparisons of pairs of groups with simultaneous confidence intervals or 2-sided tests
•When all pair of treatments are to be compared, C = t(t-1)/2
• Want the overall confidence level for all intervals to be “correct” to be 95% or the overall type I error rate for all tests to be 0.05
• For confidence intervals, construct (1-(0.05/C))100% CIs for the difference in each pair of group means (wider than 95% CIs)
• Conduct each test at =0.05/C significance level (rejection region cut-offs more extreme than when =0.05)
• Critical t-values are given in table on class website, we will use notation: t/2,C, where C=#Comparisons, = df
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Bonferroni’s Method (Most General)
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Example - Seasonal Diet Patterns in Ravens
Note: No differences were found, these calculations are only for demonstration purposes
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Comparison(i vs j) Group i Mean Group j MeanDifference1 vs 2 1.243203 1.217935 0.0252671 vs 3 1.243203 0.919375 0.3238281 vs 4 1.243203 1.161773 0.0814302 vs 3 1.217935 0.919375 0.2985602 vs 4 1.217935 1.161773 0.0561623 vs 4 0.919375 1.161773 -0.242398