| 1. |
The next output shows the structure of a data frame d
and the results of applying a one-way ANOVA model to the data.
> summary(d)
resp grp
Min. : 6.847 A:10
1st Qu.: 8.905 B:10
Median :10.063 C:10
Mean :10.095
3rd Qu.:11.000
Max. :13.033
> summary(aov(resp ~ grp, data=d))
Df Sum Sq Mean Sq F value Pr(>F)
grp ? 61.42 30.708 ???? 1.33e-07
Residuals 27 27.52 1.019
What are the values for the missing degrees of freedom (DF=) and F-statistic
(F=)?
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DF=3 and F=30.13 |
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DF=2 and F=30.13 |
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DF=2 and F=2.23 |
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Don't know. |
| 2. |
Below are results from a two-way ANOVA with factors x1
and x2, and responses collected on 100 subjects.
Df Sum Sq Mean Sq F value Pr(>F)
x1 1 1077 1077 4.893 0.029385 *
x2 1 3255 3255 14.788 0.000219 ***
x1:x2 1 1338 1338 6.081 0.015480 *
Residuals 94 20688 220
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
How many missing observations are present in the data frame?
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None. |
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2. |
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1. |
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Don't know. |
| 3. |
From the preceding ANOVA table, how would you compute the partial
effect-size measure for x2? |
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3255/20688. |
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3255/(3255+20688). |
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3255/(3255+1338+20688). |
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Don't know. |
| 4. |
Using the data set described in Question 1, but restricted to levels A and B for
factor grp, we fit a regression line to the data (20
observations). Some results are provided below:
> with(d, tapply(resp, grp, mean))
A B C
10.247899 11.765453 8.270759
> lm(resp ~ grp, data=d, subset= grp != "C")
Call:
lm(formula = resp ~ grp, data = d, subset = grp != "C")
Coefficients:
(Intercept) grpB
10.248 ?????
What is the value of the estimate for
the slope parameter? |
|
11.765. |
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1.518. |
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0.759. |
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Don't know. |
| 5. |
Would you expect to observe the same results (value of the test
statistic, and its corresponding p-value) when using a two-tailed
Student t-test vs. a simple linear regression to assess differences
between the two groups in the preceding case? |
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Yes. |
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No. |
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Don't know. |
| 6. |
Here are some data from an experiment in plant physiology, which
record the length in coded units of pea sections grown in tissue culture
with auxin present. [RR Sokal et FJ Rohlf. Biometry. 3e ed. WH Freeman
et Company, 1995] The purpose of the experiment was to test the effects
of various sugars on growth as measured by length (pea diameter measured
in ocular units, x 0.114 = mm). Four experimental
group, representing three different sugars (X2G, 2%
glucose; X2F, 2% fructose; X2S, 2% sucrose)
and one mixture of sugars (X1G1F, 1% glucose + 1%
fructose), were used, plus one control (C) without
sugar. The null hypothesis is that there is no added component due to
treatment effects among the five groups.
Data altered for the purpose of the exercise.
C X2G X2F X1G1F X2S
1 75 57 58 58 62
2 67 58 61 59 66
3 70 60 NA 58 65
4 75 59 58 61 63
5 65 62 57 57 64
6 71 60 56 NA 62
7 67 60 61 58 NA
8 67 57 60 57 NA
9 76 NA 57 57 62
10 68 61 58 59 67
Assuming the data frame peas has been converted to the long format where the
explanatory variable is now tx and the response variable
is value, the ANOVA table is shown below:
Df Sum Sq Mean Sq F value Pr(>F)
tx 4 989.6 247.41 42.37 7.13e-14 ***
Residuals 40 233.6 5.84
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
What command can be used to find the degrees of freedom for the residual sum
of squares? |
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nrow(peas)-nlevels(peas$tx) |
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length(peas$value)-nlevels(peas$tx) |
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sum(!is.na(peas$value))-nlevels(peas$tx) |
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Don't know. |
| 7. |
What command could we use to compute a 95% confidence interval for
Pearson correlation coefficient estimated from the following series of obervations?
x1 11 12 14 11 13 15 14 15 10 13 14 11 13 8 9
x2 12 13 14 11 13 16 15 16 11 14 15 12 14 8 10
|
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confint(cor(x1, x2)) |
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cor(x1, x2, conf.level=0.95) |
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cor.test(x1, x2, conf.level=0.95) |
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Don't know. |
| 8. |
In a study on cognitive performance of twenty children from four
different age groups (5, 6, 7 and 8 years), we observed the following
results with a linear regression model, considering age group as a
numerical variable:
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.7521 0.7398 1.017 0.323572
age 0.5053 0.1117 4.525 0.000299
Residual standard error: 0.5554 on 17 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.5464, Adjusted R-squared: 0.5197
F-statistic: 20.48 on 1 and 17 DF, p-value: 0.0002991
Without taking into account the missing observation, the estimated variances
are Var(x)=1.374 and Var(y)=0.642. What is the value of Pearson
coefficient of correlation between x and y? |
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0.739 |
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0.345 |
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0.592 |
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Don't know. |
| 9. |
If we were to use an ANOVA model, treating age group as a factor,
would we get the same p-value for the F-test assessing the whole model? |
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Yes. |
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No. |
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Don't know. |
| 10. |
Is the hypothesis of normality required to compute the slope of a
regression line by ordianry least squares? |
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Yes, but only that of the x-variable. |
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Yes, but only that of the y-variable. |
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Yes, both the x- and y-variable
should be normally distributed. |
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No. |
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Don't know. |