Non-Parametric Test 1.Sign test

Types of Non-Parametric Test :

the non-parametric test is Broadly divided into three categories as:

1. One-sample Test 

2.Two-sample test

3. K-sample test

In this blog we discuss the one-sample tests, 

there are four types of one-sample non-parametric test as follows: 

1. Run Test.

2.Sign Test

3. Wilcoxon Sing-Rank test

4. Kolmogorov-smirnov test.

first we discuss the sign test.

1.Sign Test

The sign test id one of the very simplest non-parametric test. the test is based on signs (i.e. plus and minus signs) hence is called sign test. the non-parametric test are alternative to the parametric one, therefore the sign test is alternative to parametric T-test, is used to test the median of population instead of mean of population. (i.e. Parametric T-test is used to testing mean of population and sign test is used to testing the median of the population they are alternative to each other when the data follows assumption of T-test we use parametric t-test, but i some situation the data dose not following the assumption of parametric test then we use non-parametric Sign test.) also it used if data is in ordinal scale.

assumptions of sign test.:

the sign test id used when the follows following assumptions.

1. the sample selected form population with  un-known median.

2. the selected variable for study is continuous.

3. the selected variable for study ids at least in ordinal scale.

Procedure of test as follows.:

let X1 ,X2, X3, ..........Xn. be a random sample of size n arranged in order of  occurrence. form the population with unknown median. now we wish to test the hypothesis that the specified value m0 (i.e. the hypothetical median is m ) of population median. 

the null and alternative hypothesis is as 

H₀: m =m_{0              for two tailed test}

V/S H₁: m ¹m₀            

the test consist following steps as

Step I: the test based on the signs then we firstly converting the all observations into the sign (means observation converted into plus and minus signs). for converting we subtract the median m0 into each observation. (i.e.  X1 - m0 ) that is we obtaining the difference for all observations from the median and observing there signs.(i.e. if difference is negative we get negative sign and difference is positive we get positive sign) in that one of the observation is equal to the median (m0 ) then we remove this observation or not consider in analysis part. then the size of sample is reduced and it is denoted as n.

Step II: now we get the Plus and minus signs then we count the number of plus sign and minus signs and it is denoted as S⁺ for total number of plus signs and S^-  for total number of minus signs.

Step III.:  Now we consider the null hypothesis is true then on the basis of postulated value of median  we expect that the value of variable is greater than median mean we get plus sign then the number of plus sign it consider as success and   number of minus sign is consider as failure approximately equal. then the distribution of sign is binomial distribution with parameter (n, p=0.5). for simplicity we consider smaller number of sign. that means the number of plus sign is less than number of minus sign then plus sign is success and minus is failure. similarly if minus signs are less than number of plus sign then minus sign is success and number of plus sign is failure.

Step IV: 

i. Small Sample Test : (i.e. n  is less than  or equal to 20 ).

if the number of observations are is less than or equal to 20 (i.e. n =<20 ).  is called small sample test.

for Decision about hypothesis we we the p-value and it is determined as 

P-value = P(S <  s)  and where s is equal to 

S = min ( S⁺,S^-)

if the number of observations are is less than or equal to 20 (i.e. n  =<20 ).  is called small sample test.

if the p-value is less than or equal to a%  level of significance then we reject the null hypothesis at a% level of significance otherwise accept the null hypothesis.

Large sample test:(i.e. n  greater than 20 )

if the number of observation is greater than 20 we use large sample test  (i.e. n  >20 ).

  for large sample test we use normal approximation for binomial distribution.

as E(S) = n*1/2 = n/2

and S.D.( S) = n*1/4= n/4

the normal approximation is z test gives as

z = (S - E(S))/S.D.(S) 

Z =  (S - (n/2))/(n/4) 

then we comparing the calculated and tabulated value of z. at  a%  level of significance.

if calculated z is less than or equal to  tabulated (critical value ) then we accept null hypothesis other wise reject the null hypothesis.

Sign Test Example:

The sign test id one of the very simplest non-parametric test. the test is based on signs (i.e. plus and minus signs) hence is called sign test. the non-parametric test are alternative to the parametric one, therefore the sign test is alternative to parametric T-test, is used to test the median of population instead of mean of population. (i.e. Parametric T-test is used to testing mean of population and sign test is used to testing the median of the population they are alternative to each other when the data follows assumption of T-test we use parametric t-test, but i some situation the data dose not following the assumption of parametric test then we use non-parametric Sign test.) also it used if data is in ordinal scale.

if these assumptions follow data set then we apply  sign test to analyse the data.:

the sign test id used when the follows following assumptions.

1. the sample selected form population with  un-known median.

2. the selected variable for study is continuous.

3. the selected variable for study ids at least in ordinal scale.

let X1 ,X2, X3, ..........Xn. be a random sample of size n arranged in order of  occurrence. form the population with unknown median. now we wish to test the hypothesis that the specified value m0 (i.e. the hypothetical median is m ) of population median. 

the null and alternative hypothesis is as 

H₀: m =m_{0              for two tailed test}

V/S H₁: m ¹m₀             

Example 1. Radom sample of 10 are given by 150, 155, 157, 178, 148, 159, 145, 147, 152, 154.

use sign test to test the average of population is 150. at 5% level of significance.

Answer: Here the distribution of population is not given, then parametric test is not used for that because assumption of normality is for parametric test is not fulfil. then we use non-parametric test.

for testing the average is 150 we use sign test for median  ( in case we consider median instead of mean) now hypothesis is

H₀: μ = 150

V/S H₁: μ ≠ 150

Firstly we converting the observation into sign because the sign test based on signs if one of the observation is equal to median we remove that observation, observation less than median we take  (-) minus sign and observation greater  than median we take (+) plus sign as follow.

    +       +       +         -             +         -          -         +            +.

first observation 150 is equal to median then it remove and 155 is greater than 150 taking (+) plus sign and  145 is less than 150 taking (-) minus sign.

now we counting the number of signs

S⁺ =Number of positive signs = 6

S^- = Number of negative signs = 3

n = total number of plus and minus signs = 9

then for  test statistic is consider S = min(S⁺, S^-) = S = min(5, 3) = 3

here the  n =9 is less than 20 then we use small sample test.

the test statistics is calculated as P-Value = P(S ≤ S) 

where S Has binomial distribution.

Note that   we consider the null hypothesis is true then on the basis of postulated value of median  we expect that the value of variable is greater than median mean we get plus sign then the number of plus sign it consider as success and   number of minus sign is consider as failure approximately equal. then the distribution of sign is binomial distribution with parameter (n, p=0.5). for simplicity we consider smaller number of sign. that means the number of plus sign is less than number of minus sign then plus sign is success and minus is failure. similarly if minus signs are less than number of plus sign then minus sign is success and number of plus sign is failure.

P-Value = P(S ≤ 3) =0.2539

The P-Value is compared with α% level of significance.

here  P-Value = 0.2539 is greater  then 5% level of significance = 0.05 hence we Accept null hypothesis, that means the given sample drawn from population having mean (Median) is 150.