Description
1. Comprehension Test
Children in a school class are given a test of comprehension of English, marked out
of 100. The children are from three different ethnic groups, which is thought to be
an important factor. The question of interest is whether there are sex differences
after allowing for ethnicity.
The data follow:
Females Males
Ethnic group E1 67 66 75 76 71 70 72 63 72 62 61 69 64 71 68 56
E2 69 57 55 63 65 55 59 47 49
E3 30 47 39 33
(a) A two-way ANOVA was run on the data, with SAS output given on pages 3
to 6. Present the results from the ANOVA following the usual Assignment
Guidelines, as given on page 1.
(b) If a one-way ANOVA is done with factor Sex, the resulting ANOVA table is:
Source DF Sum of Squares Mean Square F value p-value
Sex 1 144.166 144.166 0.99 0.3292
Error 27 3942.662 146.025
Total 28 4086.828
Briefly discuss the outcomes of the separate tests for Sex presented in parts (a)
and (b). Are the conclusions different? Give reasons to explain your answer.
SAS Output for Comprehension Test
Linear Models
The GLM Procedure
Class Level Information
Class LevelsValues
Ethnicity 3E1 E2 E3
Sex 2F M
Number of Observations Read29
Number of Observations Used29
Dependent Variable: Comprehension
Source DFSum of SquaresMean SquareF Value Pr > F
Model 5 3365.438697 673.087739 21.46<.0001
Error 23 721.388889 31.364734
Corrected Total 28 4086.827586
R-Square Coeff Var Root MSE Comprehension Mean
0.8234849.275400 5.600423 60.37931
Source DF Type I SSMean SquareF Value Pr > F
Ethnicity 23060.6400861530.320043 48.79<.0001
Sex 1 275.113176 275.113176 8.770.0070
Ethnicity*Sex 2 29.685435 14.842718 0.470.6289
SAS Output for Comprehension Test
Linear Models
The GLM Procedure
SAS Output for Comprehension Test
Lines nearly parallel, no significant interaction.
Vertical separation shows sex differences (males lower than females).
Non-zero slope shows ethnicity differences.
SAS Output for Comprehension Test
Lines nearly parallel, no significant interaction.
Vertical separation shows ethnicity differences.
Non-zero slope shows sex differences (males lower than females).
Note: E1 is the top line, E2 the middle line and E3 the lowest. (The lines are different
colours, but that doesn’t show up if viewed or printed in black and white.)
2. Invertebrates in Mussel Clumps
The following data are from Peake and Quinn (1993), Temporal variation in speciesarea curves for invertebrates in clumps of an intertidal mussel, Ecography 16, 269-
277. The two variables used in this question are:
x = log10(Area) of each of 25 mussel clumps (in dm2
), and
Y = number of different species of macroinvertebrates in each clump.
Note: Using log(Area) gives a straighter regression line than Area, which is why it
is used. This is a transformation of x, not Y ; it has been done to improve linearity,
not to stabilise variances.
The data follow. Decide if there is a useful linear relationship between x and Y , i.e.
if x is a useful linear predictor of Y .
Clump logArea Species
1 2.71 3
2 2.67 7
3 2.66 6
4 2.97 8
5 3.13 10
6 3.25 9
7 3.23 10
8 3.25 11
9 3.49 16
10 3.60 9
11 3.65 13
12 3.65 14
13 3.70 12
14 3.65 14
15 3.74 20
16 3.87 22
17 3.85 15
18 3.96 20
19 4.01 22
20 3.97 21
21 4.14 15
22 4.31 24
23 4.39 25
24 4.43 25
25 4.42 24
(a) A scatterplot of the data is given on page 8. Give comments on whether you
think the plot shows (i) linearity, (ii) constant variance.
(b) Output from a simple linear regression using logArea to predict the number
of species is given on pages 9 and 10. Present a report on this analysis that
includes (as usual) the model equation, hypotheses, assumptions, comments on
whether the analysis is valid, plus statistical conclusions and interpretation.
SAS Output for Mussel Clumps
Scatter Plot
SAS Output for Mussel Clumps
Linear Regression Results
The REG Procedure
Model: Linear_Regression_Model
Dependent Variable: Species
Number of Observations Read25
Number of Observations Used25
Analysis of Variance
Source DF
Sum of
Squares
Mean
SquareF Value Pr > F
Model 1 868.50179868.50179 117.85<.0001
Error 23 169.49821 7.36949
Corrected Total 241038.00000
Root MSE 2.71468R-Square0.8367
Dependent Mean15.00000Adj R-Sq0.8296
Coeff Var 18.09787
Parameter Estimates
Variable DF
Parameter
Estimate
Standard
Errort Value Pr > |t|
Intercept 1-25.64136 3.78287 -6.78<.0001
logArea 1 11.20214 1.03189 10.86<.0001
SAS Output for Mussel Clumps
Linear Regression Results
3. Coarse Woody Debris in Lakes
Christensen et al. (1996, Ecological Applications 6(4), 1143-1149) studied the relationships between coarse woody debris (CWD), shoreline vegetation and lake development in a sample of 16 lakes in North America.
Coarse woody debris is useful
in providing a habitat for various fish species. It is known to be related to the
riparian (river-bank, lake-edge) tree density, irrespective of whether or not humans
are present. The objective is to find out whether, after allowing for riparian tree
density, human habitation is having an effect on the CWD.
The variables below were taken around the shoreline and near-shore water:
L10CABIN = log10 of 1 + density of cabins (number km−1
),
RIP.DENS = density of riparian trees (trees km−1
), and
CWD.BASA = basal area of coarse woody debris (m2 km−1
).
LAKE AREA RIP.DENS CWD.BASA L10CABIN
Bay 69 1270 121 0
Bergner 9 1210 41 0
Crampton 24 1800 183 0
Long 8 1875 130 0
Roach 20 1300 127 0
Tenderfoot 175 2150 134 0.20412
Palmer 254 1330 65 0.462398
Street 22 964 52 0.6627578
Laura 240 961 12 0.7075702
Annabelle 85 1400 46 0.763428
Joyce 12 1280 54 0.845098
Lake hills 25 976 97 0.8864907
Towanda 58 771 1 1.10721
Black oak 234 833 4 1.1238516
Johnson 31 883 1 1.2552725
Arrowhead 40 956 4 1.40824
(a) Let Y = CWD.BASA, X1 = RIP.DENS and X2 = L10CABIN. Plots of Y
vs. X1, Y vs. X2 and X1 vs. X2 are given on page 12. Comment on any
relationships you see.
(b) SAS output for the following models is presented on pages 13 to 16. Diagnostic
graphs are shown for the last model.
i. Regression of Y on the predictor X1
ii. Regression of Y on the predictor X2
iii. Regression of Y on the two predictors X1 and X2
For each analysis above, present the model equation, hypotheses and conclusions. For the third analysis, comment on whether or not the model assumptions are satisfied.
(c) Which of the hypothesis tests from the three presented models gives the answer
to the question of interest in this situation? Explain the answer.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0 50 100 150
L10CABIN
CWD.BASA
800 1200 1600 2000
0 50 100 150
RIP.DENS
CWD.BASA
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
800 1200 1600 2000
L10CABIN
RIP.DENS
Scatterplots: CWD by L10CABIN, CWD by RIP.DENS, RIP.DENS by L10CABIN
Coarse Woody Debris SAS Output
Linear Regression Results
The REG Procedure
Model: Linear_Regression_Model
Dependent Variable: CWD.BASA
Number of Observations Read 16
Number of Observations Used 16
Analysis of Variance
Source DF
Sum of
Squares
Mean
Square F Value Pr > F
Model 1 32054 32054 24.30 0.0002
Error 14 18466 1318.96866
Corrected Total 15 50520
Root MSE 36.31761 R-Square 0.6345
Dependent Mean 67.00000 Adj R-Sq 0.6084
Coeff Var 54.20539
Parameter Estimates
Variable DF
Parameter
Estimate
Standard
Error t Value Pr > |t|
Intercept 1 -77.09908 30.60801 -2.52 0.0246
RIP.DENS 1 0.11552 0.02343 4.93 0.0002
Linear Regression Results
The REG Procedure
Model: Linear_Regression_Model
Dependent Variable: CWD.BASA
Number of Observations Read 16
Number of Observations Used 16
Analysis of Variance
Source DF
Sum of
Squares
Mean
Square F Value Pr > F
Model 1 32840 32840 26.00 0.0002
Error 14 17680 1262.86950
Corrected Total 15 50520
Root MSE 35.53688 R-Square 0.6500
Dependent Mean 67.00000 Adj R-Sq 0.6250
Coeff Var 53.04011
Parameter Estimates
Variable DF
Parameter
Estimate
Standard
Error t Value Pr > |t|
Intercept 1 121.96875 13.96871 8.73 <.0001
L10CABIN 1 -93.30142 18.29646 -5.10 0.0002
Linear Regression Results
The REG Procedure
Model: Linear_Regression_Model
Dependent Variable: CWD.BASA
Number of Observations Read 16
Number of Observations Used 16
Analysis of Variance
Source DF
Sum of
Squares
Mean
Square F Value Pr > F
Model 2 38041 19020 19.81 0.0001
Error 13 12479 959.93185
Corrected Total 15 50520
Root MSE 30.98277 R-Square 0.7530
Dependent Mean 67.00000 Adj R-Sq 0.7150
Coeff Var 46.24294
Parameter Estimates
Variable DF
Parameter
Estimate
Standard
Error t Value Pr > |t|
Intercept 1 18.16485 46.22822 0.39 0.7007
RIP.DENS 1 0.06572 0.02823 2.33 0.0367
L10CABIN 1 -56.26481 22.53059 -2.50 0.0267
4. Age of Teeth
In forensic work, scientists estimate the age of a skeleton by counting teeth cementum annulation (i.e. growth rings). Two teeth preparation methods, A and B, are
compared by estimating the ages (Y ) of twenty teeth of known age (X). The teeth
are randomly allocated to the two methods, ten to each, as follows.
Method A X = true age 49 13 38 55 44 56 7 66 18 39
Y = estimated age 50 14 38 57 44 55 7 63 20 38
Method B X = true age 51 59 32 37 12 38 4 28 58 24
Y = estimated age 51 59 29 34 10 35 5 25 57 22
A confirmatory analysis using a model with terms True Age (i.e. X), Method and
True Age×Method is required.
(a) Give the model equation for the required confirmatory analysis.
(b) SAS output from a fitted model is given on pages 18 to 20. Present a report on this analysis that includes any necessary assumptions, comments on
their validity, hypotheses, statistical conclusions at a 5% significance level, and
interpretation plus discussion.
Linear Models
The GLM Procedure
Class Level Information
Class LevelsValues
Method 2A B
Number of Observations Read20
Number of Observations Used20
Dependent Variable: Y
Source DFSum of Squares Mean SquareF Value Pr > F
Model 3 6543.6646602181.221553 946.16<.0001
Error 16 36.885340 2.305334
Corrected Total 19 6580.550000
R-SquareCoeff VarRoot MSE Y Mean
0.9943954.258997 1.51833335.65000
Source DF Type I SS Mean Square F Value Pr > F
X 16525.5352066525.5352062830.62<.0001
Method 1 15.413619 15.413619 6.69 0.0199
X*Method 1 2.715836 2.715836 1.18 0.2938
Source DF Type III SS Mean Square F Value Pr > F
X 16350.0068376350.0068372754.48<.0001
Method 1 10.463729 10.463729 4.54 0.0490
X*Method 1 2.715836 2.715836 1.18 0.2938
Data and fitted lines: Method A line (dashed) is above Method B line (solid) Data and fitted lines:
A
A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
10 20 30 40 50 60
0 10 20 30 40 50 60
True age
Estimated age