ReviewChaps3-4

8/14/2019 ReviewChaps3-4

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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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