STAT512 Homework 7

Homework 7

1)

The following example is from Snedecor and Cochran (1976, Statistical Methods, 6^th ed., Iowa State University Press, Ames, Iowa).  An experiment was conducted to study the variability of calcium concentration in turnip greens. Four (4) Plants were selected at random from a large number of plants.  From each of these plants three (3) Leaves were randomly selected and two (2) 100mg Samples were taken from each leaf. The amount of calcium was subsequently determined for each 100 mg sample.  In this problem, Sample is nested within Leaf and Leaf is nested within Plant.

In the above design, Leaf is nested within Plant (each leaf is specific to given plant) and Sample is nested within Leaf (each sample is specific to a given leaf) and also Plant.  Since Plants were selected at random from a large number of plants, Plant is a random factor.  In addition, since each Leaf was selected at random from a large number of leaves on a plant, Leaf is a random factor.  Lastly, each Sample selected from each leaf and represents one of many possible samples on a leaf.  So, Sample is random (replication).  This design is known as a two-stage nested design.

a) Assuming that Plant and Leaf are both fixed effects, complete the following ANOVA table:

b) Assuming that Plant is a fixed effect and Leaf is a random effect, complete the following ANOVA table:

c) Assuming that Plant and Leaf are both random effects, complete the following ANOVA table:

2)

Andrew Rodstrom, (PhD 2013, WSU Department of Entomology) wanted to assess a new pesticide for use with poplar tree leaf beetles.  The pesticide is applied through the irrigation water, which is fed to the trees through a drip irrigation system.

Andrew set up an experiment consisting of three fields.  Each field consisted of 9 rows of trees arranged in sets of three rows.  Three rates were of interest to the study:  full, half and none (control).  The three pesticide rates were randomly assigned to each grouping of three rows of trees within each field.  Sixteen days after pesticide application, the number of beetle larvae were counted within the first 2 meters from the ground on approximately 10 trees from each row (approximately 30 trees per treatment within each set of 3 rows of trees).

The design is a randomized complete block design (Field - 3 levels) with a one-way treatment structure (Pesticide - 3 levels).  The data for this experiment are shown in the following table:

a) Fill in the ANOVA table, including the sum of squares, degrees of freedom, mean squares and F-statistic for Pesticide Rate.

Show your computation of the sum of squares here:

b) Test the null hypothesis of equal means for the three Pesticide Rates.  Show all five parts of the test (null and research hypotheses, test statistic, rejection region and conclusion).  Use α = 0.05 for the test.

H_o:

H_a:

F =

Rejection Region:

Conclusion:

data training;

input method block response @@;

cards;

1 1 73 2 1 81 3 1 92 1 2 76 2 2 78 3 2 89 1 3 75 2 3 76 3 3 87

1 4 74 2 4 77 3 4 90 1 5 76 2 5 71 3 5 88 1 6 73 2 6 75 3 6 86

1 7 68 2 7 72 3 7 88 1 8 64 2 8 74 3 8 82 1 9 65 2 9 73 3 9 81

1 10 62 2 10 69 3 10 78

;

proc glm data=nutrition;

class method block;

model response=method block;

output out=diagnostics r=resid p=predicted;

lsmeans method/adjust=tukey lines pdiff cl;

run;

proc plot data=diagnostics;

plot resid*predicted;

proc univariate data=diagnostics normal plot;

var resid;

run;

3. The effect of cultural background on group decision making was studied by an experiment. Sixteen teams of students were formed and assigned a task. One of the response variables was the number of group interactions prior to the final group decision. Eight teams consisted of foreign students, eight of U.S. students. Half of the teams consisted of eight members, the other half of four members. Two foreign observers were used for the foreign teams, and two U.S. observers for the U.S. teams. Thus, the design may be represented as follows:

Let nationality of team be factor A, size of team factor B, and observer factor C. Which factors are crossed and which factors are nested?

The following data were collected. Factors A and B can be considered fixed and factor C random.

Create a SAS program to produce information needed for the following table.