ST 524 NCSU - Fall 2008
Homework 2 Due: Thursday September 25, 2008
I. An evaluation of sample size is carried on paper “Sample Size for Single Double and Three-Way Hybrid Corn Ear Traits”
1. Statistical Design
2. Response variables
Write down the statistical linear model for yield of grains at 13% humidity, in words if preferred.
, where
, is the yield of grains at 13% humidity for ith hybrid in jth block, i = 1, ..., 6 , j = 1, 2, 3
is the overall mean
, is the fixed effect associated with ith hybrid. ,
is the fixed effect corresponding to jth block,
is the random effect for ith hybrid and jth block.,
3. The Analysis of variance table for Yield is presented next,
How many blocks should be used if the minimum detectable difference () in any direction to be found between any two hybrids is 5 ton h-1 at a significance level of 0.05 and power equal 0.9
Use ,
4. Subsampling
Write down the statistical linear model for response Weight of 100 grains.
, where
, is the weight of 100 grains for the kth ear of ith hybrid selected in jth block, i = 1, ..., 6 , k = 1, …, 5, , j = 1, 2, 3,
is the overall mean
, is the fixed effect associated with ith hybrid. ,
is the fixed effect corresponding to jth block,
is the random effect for ith hybrid and jth block.,
is the random effect for he kth ear of ith hybrid selected in jth block,.
5. Analysis of variance table for Weight of 100 grains is presented next,
Fcalc, Hybrids = , Fcalc, Blocks =
a) Explain differences between Experimental and Sampling Error.
Experimental error represent the variation due to uncontrolled random effects associated to experimental units (plots) that received a given hybrid: plot-to-plot variation.
Sampling error represents the variation due to uncontrolled random effects associated to observational units (ears), a number of which were selected randomly within each experimental unit (plot): ear-to-ear variation.
b) Experimental Coefficient of variation (Exp. CV) is calculated by , while Am CV corresponds to . How would you interpret these two values?
24.2% represent the relative experimental variation with respect the overall mean weight of 100 grains, ie., g (experimental error) represents 24.2% of 32.4 g (mean weight) at an ear basis.
13.6% represent the relative sampling variation with respect the overall mean weight of 100 granis, ie., g (sampling error) represents 19.6% of 32.4 g (mean weight), at an ear basis.
c) Calculate the variance components in the Expected Experimental Error MS.
E(Experimental Error MS) =
E(Sampling Error MS) =
Estimated value for is 19.6 = MS(Sampling Error) =
Estimated value for is given by , thus
d) Researchers want to analyze the increase on the number of blocks and/or number of ears per plot, necessary to find significance for the set of effects similar to the observed in Table 3. What combinations of r and s would you suggest? Consider power = 0.80
Weight of 100 grains (g)Type / Hybrid / Mean / Variance / CV / Standard Deviation
Single / P30F33 / 33.2 / 35.10 / 17.8 / 5.9245
Single / P Flex / 33.5 / 12.47 / 10.5 / 3.5313
Three-way / AG 8021 / 33.8 / 17.57 / 12.4 / 4.1917
Three-way / DG 501 / 31.1 / 14.10 / 12.1 / 3.7550
Double / AG 2060 / 30.5 / 16.83 / 13.4 / 4.1024
Double / DKB 701 / 32.3 / 21.43 / 14.3 / 4.6293
Overall / 32.4 / 19.6 / 13.6 / 4.4272
Num Den
Obs Effect Label DF DF t r s alpha dendf2 fcrit ncparm power
1 hybrid hybrid 5 10 6 5 23 0.05 20 2.71089 17.0537 0.80707
2 hybrid hybrid 5 10 6 6 18 0.05 25 2.60299 16.0156 0.80255
3 hybrid hybrid 5 10 6 7 15 0.05 30 2.53355 15.5707 0.80584
4 hybrid hybrid 5 10 6 9 11 0.05 40 2.44947 14.6810 0.79937
Possible combinations: (5 blocks, 23 ears/plot), (6 blocks, 18 ears/plot), (7 blocks, 15 ears/plot), (9 blocks, 11 ears/plot)
e) Calculate the Standard Error of a hybrid mean for each case, where. Which combination of r and s in d) will result in the smallest standard error for the mean?
For original combination of r and s
.
Note that
r / s / / Power5 / 23 / / 0.80707
6 / 18 / / 0.80255
7 / 15 / / 0.80584
9 / 11 / / 0.79937
Smallest standard error of the mean is given by r = 9 and s = 11, with
f) The optimum allocation solution for number of ears selected within each plot, when minimizing the total cost, is given by, where s is the number of ears selected within each plot, c1 is the management cost for each plot and c2 is the cost per ear sampled, find the optimum allocation number of ears per plot for a relative cost c1 = 3.0 and c2 = 0.1, with and. Do any of the suggested combinations of (r, s) for the desired power correspond with the optimum number of ears to be sampled within each plot?
, optimum number of ears is 9 ears per plot.
No, there is not any combination with s = 9, but the combination of r = 9 blocks and s = 11 is the nearest to the optimum allocation of ears per plot.
g) Through independent calculations of sample size, researchers found that three blocks and 21 ears per block were adequate to attain a CV = 5%, an improvement over the observed CV average (Am CV) of 13.6%.
Comment the following
for PF0F33
Weight of 100 grains (g)Type / Hybrid / Mean / Variance / CV / Standard Deviation / Number of ears/plot (s) for desired CV=5%
Single / P30F33 / 33.2 / 35.10 / 17.8 / 5.9245 / 22
Single / P Flex / 33.5 / 12.47 / 10.5 / 3.5313 / 13
Three-way / AG 8021 / 33.8 / 17.57 / 12.4 / 4.1917 / 15
Three-way / DG 501 / 31.1 / 14.10 / 12.1 / 3.7550 / 15
Double / AG 2060 / 30.5 / 16.83 / 13.4 / 4.1024 / 17
Double / DKB 701 / 32.3 / 21.43 / 14.3 / 4.6293 / 18
Overall / 32.4 / 19.6 / 13.6 / 4.4272 / 17
Use the standard error of the mean to compare a design with three blocks and 21 ears subsampled within each plot, with a design with 7 blocks and 9 ears per plot. What power is associated with each scenario?
r / s / / Power3 / 21 / / 0.40995
7 / 9 / / 0.55294
Num Den
Obs t ro so Effect DF DF FValue ProbF Label alpha r s dendf2 ncparm fcrit power
44 6 3 5 hybrid 5 10 0.44 0.8078 hybrid 0.05 3 21 10 9.34244 3.32583 0.40995
128 6 3 5 hybrid 5 10 0.44 0.8078 hybrid 0.05 7 9 30 9.34244 2.53355 0.55294
While keeping the total number of observational units constant, an increase on the number of blocks reduces the standard error of the mean, and increases the power.
h) Calculate the sample size needed to estimate the observed difference between Single and Double Hybrids with a power = 0.80.
Single / Single / Three-way / Three-way / Double / DoubleHybrid / P30F33 / P Flex / AG 8021 / DG 501 / AG 2060 / DKB 701 / estimated contrast / Fcalc
Mean / 33.2 / 33.5 / 33.8 / 31.1 / 30.5 / 32.3
Csingle vs double / 1 / 1 / -1 / -1 / 3.9 / 0.9274
Csingle vs three-way / 1 / 1 / -1 / -1 / 1.8 / 0.1976
Note that , and
Standard Error of contrast is given by
From output,
EXP_
Obs Effect FValue Label alpha r s ncparm power Error_MS se_contrast
1 contrast 0.93 Single vs Dual 0.05 6 23 8.53244 0.80158 212.34 1.24044
2 contrast 0.93 Single vs Dual 0.05 7 19 8.22329 0.79241 178.82 1.15953
3 contrast 0.93 Single vs Dual 0.05 8 17 8.40878 0.80495 162.06 1.09161
4 contrast 0.93 Single vs Dual 0.05 9 15 8.34695 0.80487 145.30 1.03745
5 contrast 0.93 Single vs Dual 0.05 10 13 8.03780 0.79223 128.54 0.99437
i) Calculate the sample size needed to estimate the observed difference between Single and Three-way Hybrids with a power = 0.80.
For Contrast “single vs Three-way”, the power for any of the combinations of r and s considered is below 0.80, the largest values for power is given by
Obs Effect FValue Label alpha r s ncparm power
215 contrast 0.20 Single vs Triple 0.05 10 24 3.16098 0.41304
216 contrast 0.20 Single vs Triple 0.05 10 25 3.29268 0.42705
Increasing r and s in calculation of power, we get power = 0.8 for the following combination of r and s
Obs Label FValue effect alpha r s ncparm power
585 Single vs Triple 0.20 c 0.05 7 91 8.38976 0.80031
672 Single vs Triple 0.20 c 0.05 8 79 8.32390 0.80100
1520 Single vs Triple 0.20 c 0.05 17 36 8.06049 0.80092
1714 Single vs Triple 0.20 c 0.05 19 32 8.00780 0.79941
j) Does any of the suggested sample size agree with the needed number of blocks found in 3?
No, since the minimum detectable difference between a pair of means is larger than any difference observed between mean, pairwise-difference range is 0.1 to 3.8 in absolute value.
Tuesday September 15, 2008 5