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Review Chaps 3-4
Chapter 3
A Closer Look at Assumptions
Robustness: a statistical procedure is robustto departures from a particular assumption ifit is valideven when the assumption is not met. Validmeans that the uncertaintymeasures, such as confidence intervals and p-values, are very nearly equal to those under
the assumption. The t procedures assume that the data are normal, that the two
populations have equal variances, and that all the observations are independent of one
another.
Departures from the Normality Assumption: The tprocedures are robust to departures
from normality. Data depart from normality when their distribution is not symmetrical,
bell- or mound-shaped, or when the tails are either too long or short. While this issubjective to some extent, assessing how severe is the departure from normality is an
important part of your training.
Departures from the Equal Variances Assumption: These departures can be moreserious. This condition is best checked by looking at histograms of both samples as well
as the sample standard deviations. Often, so long as the sample sizes are similar the
uncertainty measures will still be reasonably close to the true ones.
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Departures from Independence: This can be caused by serial or spatial correlation or by
cluster effects. These assumptions can usually be easily checked by considering the
experimental design and data collecting procedures. Data that fail to meet theindependence assumption require methods other than those presented here.
What are some examples of data sets that violate the independence assumption?
Resistance: A statistical procedure is resistant if it does not change very much when asmall part of the data changes. The t procedures are not resistant to outliers. Outliers
should be identified and the analysis performed with and without the outlier and the
results reported in the published results of the experiment.
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Transformations of the Data: sometimes departures from normality can be corrected bytransformations. The most common transformation is the log transform. There are two
common log transformations. It is common for writers to use log to mean both the natural
log and the log base 10. In my notes, I will often use ln to mean natural log and log tomean log 10, or log to mean either natural or log 10. Your text will use log to mean
natural log and will use log10 to mean log 10, though the use oflog10 will be rare. Pleasedo not let this confuse or upset you.
Multiplicative Treatment Effect under the log Transformation: If an additive effect is
found for log transformed data, i.e.,
log( )y x = + ,
Then, to get back to the original scale,
( )log( )x
ye e xe
+
= = Thus, e
is the multiplicative effect on the original scale of measurement. To test for a
treatment effect, we use the usual t-test on the log-transformed data.
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Population Model: Estimating the Ratio of Population Medians
When drawing inferences about population means using a log or natural log transform,
there is a problem with transforming back to the original scale of the data because themean of the log transformed sample is not the log of the mean of the original. So taking
the antilog does not give an estimate of the mean on the original scale.
However, if the log transformed data have a symmetric distribution then
mean[log(Y)] = median[log(Y)]
and
median[log(Y)] = log[median(Y)].
In words, The median, or 50th percentile, of the log transformed values is the log of the
50
th
percentile of the original values. So, when we transform back to the original scale,we are now drawing inferences about the median.
If we denote the averages of the log transformed means as 1 2,Z Z then the difference of
these two quantities estimates the log of the ratio of their medians. That is,
2 1Z Z =
( )
( )2
1
medianlog
median
Y
Ywhere Y1 and Y2 represent the two samples and hence the
right-hand side is an estimate of the log of the ratio of the two population medians.
Other transformations: There are many transformations one can try. There are rules ofthumb, but it usually boils down to trial and error. Some common transformations are
square root, reciprocal, and the arcsine.
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Wilcoxon Sum Rank Test
Say you believe that students who go home to their families for Thanksgiving Weekend
actually do better on their exams because they need to decompress more than they needto study. Say you took a random sample of 8 students who went home for Thanksgiving
and 8 who stayed in Missoula and studied, and then obtained their final exam scores outof a total of 200 possible.
Here are the resulting data
Went Home Studied
113.2595.94
90.04
104.44119.21
106.88
94.99131.09
137.9134142.4956
129.6706
115.4934183.4077
123.5596
94.7618102.0240
The hypothesis test is then:
H0: h-s=0 versus Ha: h-s>0
To perform the sum-rank test, we need to rank the two samples together and find the
ranks. Once we have found the ranks, the test statistic can be calculated.
Under the null hypothesis of no difference, we have( )( )
( )1 1 21 1,var
2 12W
n N n n N W
+ += =
Our two samples are just large enough (each >7) so we can perform this test by
calculating a z-score.
( )
wW
zSD W
= .
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HomeRanked
StudiedRanked
Ranks
90.0400
94.990095.9400
104.4400
106.8800
113.2500
119.2100
131.0900
94.7618
102.0240
115.4934
123.5596
129.6706
137.9134142.4956
183.4077
12
345
6
78
9
10
1112
13
1415
16
We now sum the ranks for the home data. This yields W=1+3+4+6+7+8+10+12=51.
For this example,
51 680.1992
85.33z
= =
So, we fail to reject the null hypothesis of no difference between the two groups.
Wilcoxon Signed-Rank Test for Paired Data:
To compute the signed-rank test statistic we perform the following calculations:
1) Compute the difference in each of the n pairs.2) Drop the zeros from the list.3) Order the absolute differences from smallest to largest and assign ranks.4) The signed rank statistic, S, is the sum of the ranks from the pairs for which the
difference is positive.
This procedure can be performed in SPSS under non-parametric tests2 relatedsampleschoose Wilcoxon.
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Exact p-value for the Signed-Rank Test: The exact p-value is the proportion of allassignments of outcomes within each pair that lead to a test statistic as least as extreme as
the observed. In the schizophrenia example, we assign affected or not affected to
each pair, regardless of true status, in 215
different ways. Then calculate the statistic underthese different assignments. The distribution of the statistics calculated in this way is then
the true distribution under the null hypothesis of no treatment effect.
Normal approximation for the Signed-Rank Test: A normal approximation give us an
expected value and standard deviation:
s=n(n+1)/4
SD(S)=[n(n+1)(2n+1)/24]1/2
We then compare the usual z-statistic to the quantiles of the normal distribution to obtain
a p-value in the usual way.
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