2sign test & mcnemar test

7/28/2019 2Sign Test & McNemar Test

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Sign Test & McNemar TestKAREN FATIMA R. MACALIMBON


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Sign Test

Frequently used nonparametric test thatdoes not depend on the assumptions ofthe t test, or measurement beyond theordinal scale.

FOCUSES on the MEDIAN rather than themean as a measure of central tendency orlocation.

Only assumption underlying the test: thedistribution of the variable of interest iscontinuous.


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The sign test gets its

name from the fact thatpluses and minusesrather than numericalvalues, provide the raw

data used in thecalculations.


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Sign test for two Independentsamples (Median two sample case)

Median test. Used to compare themedian of two independent samples.

Counterpart of the t-test underparametric

The data consist of two independentsamples of n1 and n2 observations.The medians of the samples aretaken jointly.


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In each sample observation,

the values above (>) the medians= plus (+) sign

those at or below (


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Formula

2 =

= Chi square test

a and c = observed (+) frequencies

b and d = observed (-) frequencies

k and l = the row total

m and n = the column total

N = the grand total


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example

Consider the test scores of 12 female and 9 male students ina spelling test.

I. Problem: Is there a significant difference in the performanceof two groups?

II. Hypothesis:Ho: There is no significant difference in the

performance of the two groups.

Ha: There is a significant difference in theperformance of the two groups.

Female 12 26 25 10 10 10 22 20 19 17 17 15

Male 6 22 19 7 8 12 16 8 19


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III. Level of Significance:

= .05

df = (c-1)(v-1).05 = 3.841

IV. Statistics: Median test fortwo independent samples.


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Computations: Median = 16.

- + + - - - + + + + + -

Female 12 26 25 10 10 10 22 20 19 17 17 15

Male 6 22 19 7 8 12 16 8 19

- + + - - - - - +

+ - Total

F a 7 b 5 k 12

M c 3 d 6 l 9

Total m 10 n 11 N 21

=


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V. Decision Rule: If

computed> tabular, Reject Ho.

VI. Conclusion:

computed = 1.288 < tabular3.841 at .05 level of significance,with 1 df, the Ho is confirmed.


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Sign test for two-correlated

samples (Fisher Sign Test)

Counterpart of t-test forcorrelated sample

Compares two correlated

samples and is applicable todata composed of N pairedobservations.


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Difference between each pair ofobservation is obtained.

This test is based on the idea that half thedifference between the paired observationswill be positive abd the other half will benegative.

Formula:

z =

z = the Fisher Sign Test

D = the difference between the number of+ & - signs


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Example

Consider the pretest and posttest results before and

after the implementation of the program

Pretest Posttest

x y

15 19

19 30

31 26

36 8

10 10

11 619 17

15 13

10 22

16 8


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I. Problem: Is there a significant difference

between the pretest and posttestresults of the 10 students?

II. Hypothesis:

Ho: There is no significant differencebetween the pretest and posttestresults of the 10 students.

Ha: There is a significant differencebetween the pretest and posttestresults of the 10 students.

III. Level of Significance:

= .05 .0 = 1.96

IV. Statistics: z test (the Fisher Sign Test)


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Computation:

Pretest Posttest Sign of x-y

x y D

15 19 -

19 30 -

31 26 +

36 8 +

10 10 0

11 6 +

19 17 +15 13 +

10 22 -

16 8 +

z =


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V. Decision Rule:

if z computed > z tabular= Reject Ho

VI. Conclusion:

z computed= .67 < ztabular 1.96 at .05 level ofsignificance, Accept Ho.


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Sign test for K independentsamples (The Median test: Multi-

Sample test) A straightforward extension of the median

test for two independent samples.

The test is used for k independentsamples.

= ( )

Where: = chi square test

O = observed frequencies

E = expected frequencies


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example

A sampling of the acidity of rain for tenrandomly selected rainfalls was recordedat three different locations in the provinceof Northern Samar, Biri Island, Catarmanand Silvino Lubos. The pH readings forthese 30 rainfalls are shown in the table.

(Note that the pH readings range from 0to 14; 0 is acid, 14 is alkaline. Pure waterfalling through clean air has a pH readingof 5.7)


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Use the median test at .05 level of significance to testHo that there is no significant difference among the pHreading of the three different municipalities of NorthernSamar.

Biri Catarman Silvino Lobos

4.4 4.6 4.7

4.0 4.5 4.8

4.1 4.3 5.0

3.5 3.8 4.9

2.4 4.2 3.9

3.8 4.5 4.5

4.2 4.7 4.6

3.9 4.3 4.3

4.1 4.5 4.0

4.2 4.8 4.7


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I. Problem: Is there a significant differencein the pH readings among thethree different municipalities ofNorthern Samar?

II.Hypothesis:III.Level of Significance: = .05

df = (c-1)(r-1) = 2

.05 = 5.991

IV. Statistics: Median test for k independentsamples


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Computation

Biri CatarmanSilvinoLobos

4.4 + 4.6 + 4.7 +

4.0 - 4.5 + 4.8 +

4.1 - 4.3 - 5.0 +

3.5 - 3.8 - 4.9 +

2.4 - 4.2 - 3.9 -

3.8 - 4.5 + 4.5 +

4.2 - 4.7 + 4.6 +

3.9 - 4.3 - 4.3 -

4.1 - 4.5 + 4.0 -

4.2 - 4.8 + 4.7 +


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MunicipalitiesAbove

4.3At or

below 4.3Total

O E O E

Biri 1 (4.7) 9 (5.3) 10

Catarman 6 (4.7) 4 (5.3) 10

Silvino Lubos 7 (4.7) 3 (5.3) 10

Total 14 16 30

Solve for expected frequencies:

1 10

0

= 4.716 10

0

= 5.3

= ( )

=8.296


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V. Decision Rule:

if computed > tabular,Reject Ho.

VI. Conclusion:

The computed of 8.296 >

tabular value of 5.991 at.05 level of significance with2 df, hence Reject Ho,Accept Ha


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Mcnemar

It is named after Quinn McNemar, whointroduced it in 1947.

Used to determine whether there isevidence of a difference between theproportions of two related samples.

Uses a test statistic that approximatelyfollows the normal distribution enablingyou to carry out either a one-tail or two-

tail test.


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Is a chi square test for

situations when samples arematched, that is, they are notindependent.

This is a before and afterdesign which is tested to find

out whether there is asignificant change between thebefore and after situations.


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2x2 contingency table forMcNemar Test

A = # of respondents that answer YES to condition 1 andYES to condition 2

B = # of respondents that answer YES to condition 1 andNO to condition 2

C = # of respondents that answer NO to condition 1 andYES to condition 2

D = # of respondents that answer NO to condition 1 andNO to condition 2

n= # of respondents in the sample

Condition (group 2)

Condition

(group 1) Yes No Row totalYes a b a + bNo c d c + d

Column

total

a + c b + d n


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Sample proportions of interest: P1 = A + B / n = proportion of respondents who

answer YES to condition 1

P2 = A + C /n = proportion of respondents whoanswer YES to condition 2

Population proportions of interest:

1 = proportion of population who would answer YESto condition 1

2 = proportion of population who would answer YESto condition 2


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Example

Actually purchased

Planned topurchase

Yes No Total

Yes 200 50 250

No 100 650 750

Total 300 700 1,000

You want to determine whether there is a difference

between the population proportion who planned topurchase a big screen tv and the population whoactually purchased.

Ho: 1 = 2Ha: 1 2


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Using 0.05 level ofsignificance, the criticalvalues are -1.96 and +1.96and the decision rule is:

Reject Ho if Z < -1.96 orif Z > +1.96


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A = 200 B = 50

C = 100 D = 650

P1 = A+B/n

P2 = A+C/n

Z = B-C/ B+C


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Thank You!

2sign test & mcnemar test

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