100 STATISTICAL TESTS phần 4 doc

The null hypothesis of equal variances is rejected if B is larger than the critical value... If the observed ratio exceedsthis critical value, the null hypothesis of equal variances shou

Test 29 The Link–Wallace test for multiple

comparison of K population means

(equal sample sizes)

Object

To investigate the significance of all possible differences between K population means.

Limitations

1 The K populations are normally distributed with equal variances.

2 The K samples each contain n observations.

where w i (x) is the range of the x values for the ith sample

w( ¯x) is the range of the sample means

n is the sample size.

The null hypothesis µ1 = µ2 = · · · = µ K is rejected if the observed value of K Lislarger than the critical value obtained from Table 10

Example

Three advertising theme tunes are compared using three panels to assess their pleasure,

using a set of scales The test statistic D is computed as 2.51, and then the three

differences between the ranges of means are also calculated Tunes 3 and 2 differ as dotunes 3 and 1, but tunes 1 and 2 do not

Using K8,3;0.05 = 1.18, the critical value for the sample mean differences is

Test 30 Dunnett’s test for comparing K treatments

where S02is the sum of squares of deviations from the group mean for the control group

and S2j is a similar expression for the jth treatment group The standard deviation of the

differences between treatment means and control means is then

S( ¯d)="2S W2/n

The quotients

D j = ¯x j − ¯x0

S( ¯d) ( j = 1, 2, , K)

are found and compared with the critical values of |D j| found from Table 11 If

an observed value is larger than the tabulated value, one may conclude that the

corresponding difference in means between treatment j and the control is significant.

Example

Four new topical treatments for athlete’s foot are compared with a control, which isthe current accepted treatment Patients are randomly allocated to each treatment andthe number of days to clear up of the condition is the treatment outcome Do the new

2× 74.68110

Test 31 Bartlett’s test for equality of K variances

Samples are drawn from each of the populations Let s2j denote the variance of a sample

of n j items from the jth population ( j = 1, , K) The overall variance is defined by

B will approximate to a χ2-distribution with K − 1 degrees of freedom The null

hypothesis of equal variances is rejected if B is larger than the critical value.

In a subsequent test of equality of variances the engineer takes smaller samples whichnecessitate his use of tables for comparison In this case his sample values produce an

M value which is less than the tabulated value of 9.21 from Table 12 So again he

concludes that the variances are the same

Critical value χ3;0.052 = 7.81 [Table 5]

Hence the null hypothesis is not rejected

M= 2.30259{9 log 4.11 − 2 log 6.33 − 2 log 1.33 − 2 log 4.33 − 3 log 4.33}

The critical value of M for α = 0.05, K = 4 is 9.21 [Table 12]; even for C = 2.0 Do

not reject the null hypothesis

Test 32 Hartley’s test for equality of K variances

Object

To investigate the significance of the differences between the variances of K normally

distributed populations

Limitations

1 The populations should be normally distributed

2 The sizes of the K samples should be (approximately) equal.

Method

Samples are drawn from each of the populations The test statistic is Fmax= s2

max/s2minwhere s2maxis the largest of the K sample variances and s2minis the smallest of the K

sample variances

Critical values of Fmaxcan be obtained from Table 13 If the observed ratio exceedsthis critical value, the null hypothesis of equal variances should be rejected

Example

Four types of spring are tested for their response to a fixed weight since they are used

to calibrate a safety shut-off device It is important that the variability of responses

is equal Samples of responses to a weight on each spring are taken The Hartley F

statistic is calculated to be 2.59 and is compared with the critical tabulated value of2.61 [Table 13] Since the calculated statistic is less than the tabulated value the nullhypothesis of equal variances is accepted

Test 33 The w /s-test for normality of a population

Object

To investigate the significance of the difference between a frequency distribution based

on a given sample and a normal frequency distribution

Limitations

This test is applicable if the sample is taken from a population with continuousdistribution

Method

This is a much simpler test than Fisher’s cumulant test (Test 20) The sample standard

deviation (s) and the range (w) are first determined Then the Studentized range q = w/s

We have two samples for consideration They are taken from two fluid injection

processes The two test statistics, q1and q2, are both within their critical values Hence

we accept the null hypothesis that both samples could have been taken from normaldistributions Such tests are particularly relevant to quality control situations

for n1= 4, 1.93 and 2.44 [Table 14];

for n2= 9, 2.51 and 3.63 [Table 14]

Hence the null hypothesis cannot be rejected

Test 34 Cochran’s test for variance outliers

Object

To investigate the significance of the difference between one rather large variance and

K− 1 other variances

Limitations

1 It is assumed that the K samples are taken from normally distributed populations.

2 Each sample is of equal size

Method

The test statistic is

C= largest of the s2i sum of all s2i where s2i denotes the variance of the ith sample Critical values of C are available from

Table 15 The null hypothesis that the large variance does not differ significantly from

the others is rejected if the observed value of C exceeds the critical value.

Example

In a test for the equality of k means (analysis of variance) it is assumed that the k

populations have equal variances In this situation a quality control inspector suspectsthat errors in data recording have led to one variance being larger than expected Sheperforms this test to see if her suspicions are well founded and, therefore, if she needs

to repeat sampling for this population (a machine process line) Her test statistic, C=0.302 and the 5 per cent critical value from Table 15 is 0.4241 Since the test statistic

is less than the critical value she has no need to suspect data collection error since thelargest variance is not statistically different from the others

Critical value C9;0.05= 0.4241 [Table 15]

The calculated value is less than the critical value

Do not reject the null hypothesis

Test 35 The Kolmogorov–Smirnov test for goodness

of fit

Object

To investigate the significance of the difference between an observed distribution and

a specified population distribution

Limitations

This test is applicable when the population distribution function is continuous

Method

From the sample, the cumulative distribution S n (x)is determined and plotted as a step

function The cumulative distribution F(x) of the assumed population is also plotted

on the same diagram

The maximum difference between these two distributions

His test statistic, maximum D is 0.332 and the 5 per cent critical value from

Table 16 is 0.028 So he cannot assume such an arrival distribution and must seekanother to use in his traffic model Without a good distributional fit his traffic modelwould not produce robust predictions of flow

Test 36 The Kolmogorov–Smirnov test for comparing two populations

Given samples of size n1 and n2 from the two populations, the cumulative

distribu-tion funcdistribu-tions S n1(x) and S n2(y)can be determined and plotted Hence the maximumvalue of the difference between the plots can be found and compared with a criticalvalue obtained from Table 16 If the observed value exceeds the critical value the nullhypothesis that the two population distributions are identical is rejected

Test 37 The χ2-test for goodness of fit

2 The division into classes must be the same for both distributions

3 The expected frequency in each class should be at least 5

4 The observed frequencies are assumed to be obtained by random sampling

of the K classes This statistic is compared with a value obtained from χ2tables with

ν degrees of freedom In general, ν = K − 1 However, if the theoretical distribution contains m parameters to be estimated from the observed data, then ν becomes K −1−m.

For example, to fit data to a normal distribution may require the estimation of the mean

and variance from the observed data In this case ν would become K− 1 − 2

If χ2is greater than the critical value we reject the null hypothesis that the observedand theoretical distributions agree

Example

The experiment of throwing a die can be regarded as a general application and numerousspecific situations can be postulated In this case we have six classes or outcomes, each

of which is equally likely to occur The outcomes could be time (10 minute intervals)

of the hour for an event to occur, e.g failure of a component, choice of supermarketcheckout by customers, consumer preferences for one of six types of beer, etc Sincethe calculated value is less than the critical value from the table it can be reasonablyassumed that the die is not biased towards any side or number That is, it is a fair die.The outcomes are equally likely

Numerical calculation

A die is thrown 120 times Denote the observed number of occurrences of i by O i,

i = 1, , 6 Can we consider the die to be fair at the 5 per cent level of significance?

Critical value χ5;0.052 = 11.1 [Table 5].

The calculated value is less than the critical value

Hence there are no indications that the die is not fair

Test 38 The χ2-test for compatibility of K counts

Method (a) Times for counts are all equal

Let the ith count be denoted by N i Then the null hypothesis is that N i= constant, for

i = 1, , K The test statistic is

where ¯N is the mean of the K counts K

i=1N i /K This is compared with a value

obtained from χ2tables with K − 1 degrees of freedom If χ2 exceeds this value thenull hypothesis is rejected

Method (b) Times for counts are not all equal

Let the time to obtain the ith count be t i The test statistic becomes

A lime producing rotary kiln is operated on a shift-based regime with four shift workers

A training system is to be adopted and it is desired to have some idea about how theworkers operate in terms of out-of-control warning alerts Do all the four shift workersoperate the kiln in a similar way? The number of alerts over a full shift is recorded andthe test statistic is calculated Do the four workers operate the kiln in a similar way? Theanswer to this question has bearing on the sort of training that could be implementedfor the workers

The chi-squared test statistic is 13.6 and the 5 per cent critical value of the chi-squareddistribution from Table 5 is 7.81 Since the calculated chi-squared value exceeds the

5 per cent critical value we reject the null hypothesis that the counts are effectively

Numerical calculation

N1= 5, N2= 12, N3= 18, N4= 19, ¯N = 11, ν = K − 1 = 4 − 1 = 3

Using method (a), χ2= 13.6

The critical value χ3;0.052 = 7.81 [Table 5]

Hence reject the null hypothesis The four counts are not consistent with each other

Test 39 Fisher’s exact test for consistency in a 2 × 2 table

Method

A 2× 2 contingency table is built up as follows:

where the summation is over all possible 2× 2 schemes with a cell frequency equal

to or smaller than the smallest experimental frequency (keeping the row and columntotals fixed as above)

If

p is less than the significance level chosen, we may reject the null hypothesis of

independence between samples and classes, i.e that the two samples have been drawnfrom one common population

Example

A medical officer has data from two groups of potential airline pilot recruits Twodifferent reaction tests have been used in selection of the potential recruits He uses amore sophisticated test and finds the results given for the two samples Class 1 repre-sents quick reactions and class 2 represents less speedy reactions Is there a differencebetween the two selection tests?

Since some cell frequencies are less than 5 the medical officer uses Fisher’s exact

Test 40 The χ2-test for consistency in a 2 × 2 tableObject

To investigate the significance of the differences between observed frequencies for twodichotomous distributions

Limitations

It is necessary that the two sample sizes are large enough This condition is assumed

to be satisfied if the total frequency is n > 20 and if all the cell frequencies are greater

than 3 When continuous distributions are applied to discrete values one has to applyYates’ correction for small sample sizes

χ2exceeds the critical value the null hypothesis of independence between samples andclasses is rejected In other words, the two samples were not drawn from one commonpopulation

Numerical calculation

a = 15, b = 85, c = 4, d = 77

a + b = 100, c + d = 81, a + c = 19, b + d = 162

Test 41 The χ2-test for consistency in a K × 2 tableObject

To investigate the significance of the differences between K observed frequency

distributions with a dichotomous classification

Limitations

It is necessary that the K sample sizes are large enough This is usually assumed to be

satisfied if the cell frequencies are equal to 5

−402

80 = 5.495Hence do not reject the null hypothesis

Test 42 The Cochran test for consistency in an n × K

table of dichotomous data

Object

To investigate the significance of the differences between K treatments on the same n

elements with a binomial distribution

Limitations

1 It is assumed that there are K series of observations on the same n elements.

2 The observations are dichotomous and the observations in the two classes arerepresented by 0 or 1

3 The number of elements must be sufficiently large – say, greater than 10

Method

From the n × K table, let R i denote the row totals (i = 1, , n) and C j denote the

column totals ( j = 1, , K) Let S denote the total score, i.e S = i R i = j C j.The test statistic is

Q=

K(K − 1)

j (C j − ¯C)2

This approximately follows a χ2-distribution with K− 1 degrees of freedom

The null hypothesis that the K samples come from one common dichotomous distribution is rejected if Q is larger than the tabulated value.

Example

A panel of expert judges assess whether each of four book cover formats is acceptable

or not Each book cover format, therefore, receives an acceptability score The Cochran

Q statistic is calculated as 12.51, which is larger than the tabulated chi-squared value of

7.81 [Table 5] It seems that the book covers are not equally acceptable to the judges

Test 43 The χ2-test for consistency in a 2 × K table

Object

To investigate the significance of the differences between two distributions based on

two samples spread over K classes.

Limitations

1 The two samples are sufficiently large

2 The K classes when put together form a complete series.

e ij= N i n ·j

N1+ N2

.The test statistic is

This is compared with the value obtained from a χ2table with K−1 degrees of freedom

If χ2exceeds this critical value, the null hypothesis that the two samples originate fromtwo populations with the same distribution is rejected

Example

Our medical officer from Tests 39 and 40 now wishes to have a third (or middle)class which represents a reserve list of potential recruits who do not quite satisfy thestringent class1 requirements She can still use the chi-squared test to compare the