{"id":385,"date":"2019-08-02T08:46:56","date_gmt":"2019-08-02T08:46:56","guid":{"rendered":"http:\/\/hsp16.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=385"},"modified":"2019-08-02T08:49:21","modified_gmt":"2019-08-02T08:49:21","slug":"non-parametric-test-ii-counts-or-percent-chi-square-mc-nemar-test-cocharn-q-test-spearman-correlation","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/hsp16\/chapter\/non-parametric-test-ii-counts-or-percent-chi-square-mc-nemar-test-cocharn-q-test-spearman-correlation\/","title":{"rendered":"Non \u2013parametric test II \u2013 counts or percent, chi square, Mc Nemar test, Cocharn Q test, Spearman correlation."},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n
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Introduction<\/strong><\/p>\r\n \r\n

The real life situations, there are a number of occasions when the universe distribution is not normal. Hence to test hypothesis relating to non-normal populations, there is a necessity to develop a different group of tests which are not based on the too restrictive assumptions. Such tests which are free from distribution are called Non-Parametric tests. In addition to the above, chi-square test enables one to find out the significance of difference between two or more population proportions. When the dependent variable samples are continuous in nature, then the sign and Wilcoxon tests are appropriate for two dependent sample studies. Count means determine the total number of collected items. Sometimes the investigator may be interested in finding out whether to qualitative characters tend to occur together in a given population.<\/p>\r\n \r\n

Non-Parametric Test<\/strong><\/p>\r\n \r\n

These non-parametric tests can always address situations where the data are nominal in nature like increase, decrease, no change, or ordinal like ranked. The non-parametric tests in that sense are now strengthening the hands of statisticians to analyzer and explain and type of situation distribution based or distribution free. As regards the historical background about non-parametric tests, it could be stated that the Chi-square test developed by karl Pearson in 1900 was the earliest of non-parametric tests formulated. This was followed by the contributions of several statisticians like spearman, with rank correlation tests, which became popular after Harold Hotelling who used the rank correlation coefficients in 1936. It was in 1945 when wilcozxon developed a test for two sample cases, the real growth of non-parametric tests started taking place.<\/p>\r\n \r\n\r\nAdvantages in using the non-parametric tests are<\/strong>:\r\n

    \r\n \t
  1. They are not dependent on the nature of distribution of the population.<\/li>\r\n \t
  2. They are simple to understand and easy to apply\/<\/li>\r\n \t
  3. They are less time consuming and once the result is statistically significant, no further ork is necessary.<\/span><\/li>\r\n \t
  4. They are very useful to researchers where the sample size is very small and adequate data from samples cannot be compiled.<\/span><\/li>\r\n \t
  5. They are amenable for application for any type of data \u2013qualitative, quantitative or ranked. f)They are not based on very restrictive assumptions and in the case of parametric tests.<\/span><\/li>\r\n<\/ol>\r\n

    Chi square test<\/strong><\/p>\r\n \r\n

    Karl Pearson in 1900 developed a statistic procedure for testing the significance of discrepancy between experimental values and theoretical values obtained under some theory or hypothesis. This test is known as <\/span>\u1d61<\/strong>2<\/strong> (chi-square) test. It is a test of goodness of fit and is used to find whether the deviation between observation and theory may be attributable to change or any other factor. Chi-square test is applicable to enumeration data and not to data collected on measurements on continuous scale .<\/span><\/p>\r\n \r\n

    Data applicable to Chi square Test<\/strong><\/p>\r\n \r\n

    1 Qualitative or categorical variables<\/strong><\/p>\r\n \r\n

    There are a number of qualitative variables inhuman populations such as colour of the skin, eye, hair, etc. The individuals possessing these characters are counted and the number of occurrences in the sample is given in the form of frequency. These characteristics are expected to occur in certain frequencies, called as expected frequencies. Similarly, there are a number of categorical variables such as blood group education level, employment level, etc. These variables are categorized and the number of individuals possessing these characters is counted and the numbers of occurrences in the sample is given in the form of frequency against each category.<\/span><\/p>\r\n \r\n

    For example number of persons with the same blood group are counted and given against each category like A,B.AB and O.<\/span><\/p>\r\n \r\n

    ii) Qualitative or categorical variables given in contingency table<\/strong><\/p>\r\n \r\n

    Sometimes the investigator may be interested in finding out whether to qualitative characters tend to occur together in a given population. In such a case, the individuals are counted for the presence or absence of to specific characters and are given in the form of a contingency table 1<\/span><\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n \r\n\r\nTable 1\r\n\r\n\"\"<\/a>\r\n\r\n \r\n

    The table 1 presented in the form of a table with variables, one each in row and column and under each category the observations are placed in two categories. This table is a 2x2 contingency table, since data are presented in two rows and to columns. Depending upon the number o categories for each variable the number of rows and columns vary. iii) Qualitative or categorical variables in the form of proportions<\/p>\r\n \r\n

    A demographer may be interested in finding out whether the proportion of material status of men in four different cities is the same. He counts the number of persons \u2013 married and single and presents it in the form of proportions like married \/ Total and single \/total against each city.<\/p>\r\n \r\n

    Conditions for validity of Chi-square test Chi-square test can be used only if the following conditions are satisfied<\/p>\r\n\r\n

      \r\n \t
    1. \u2018N\u2019 the total frequency should be reasonably large, say greater than 50.<\/li>\r\n \t
    2. the observations in the sample should be independent. This means no individual item should be included twice or more in the sample.<\/li>\r\n \t
    3. No theoretical frequency should be small. Preferably, each theoretical frequency should be larger than 10 but should not be less than 5.<\/li>\r\n \t
    4. The data should be given in original unit<\/li>\r\n<\/ol>\r\n

      Application of Chi-square Test<\/strong><\/p>\r\n \r\n

      i) Chi \u2013square test of goodness of fit or chi \u2013square test with a priori hypothesis.<\/strong><\/p>\r\n \r\n

      Goodness of fit is a generic term used to indicate how far an observed frequency distribution fits ell with the expected frequency distribution based on some theory or expectation. Since the expected frequencies are calculated based on some theory or hypothesis, it is also called Chi \u2013 square test with a priori hypothesis.<\/span><\/p>\r\n \r\n

      In this case the null hypothesis(H<\/strong>0<\/strong>) <\/strong>is \u201c there is no difference between observed frequency and expected frequency\u201d . The alternate hypothesis (HA) is \u201cthere is difference between observed frequency and expected frequency.<\/span><\/p>\r\n \r\n

      ii)Chi Square test for association between attributes or independence of attributes or chi-square test without a priori hypothesis. It is used to find whether two characters have the tendency to remain together or remain independently. For example, whether a patient suffering from a disease is cured because of the drug or the recovery from illness is independent of the drug. In this case, the frequencies are given with the form of a contingency table.<\/span><\/p>\r\n \r\n

      Hence, the null hypothesis (H0) is \u201c There is no association between two characters\u201d. i.e.,the drug id not effective in curing the disease. The alternate hypothesis is \u201cThere is association between the two characters\u201d. i.e., the drug is effective in curing the disease.<\/span><\/p>\r\n \r\n

      iii)\u00a0\u00a0 <\/strong>Chi-square test to test the equality of proportions.<\/strong><\/p>\r\n \r\n

      As stated if an investigator is interested in finding out whether the proportion of married men is same in four major cities. The null hypothesis (H0) is \u201cThe proportion of married men is the same in four major cities. The alternate hypothesis (HA) is \u201c The proportion of married men is not the same in four major cities.<\/span><\/p>\r\n \r\n

      iv) Chi-square test of homogeneity<\/strong><\/p>\r\n \r\n

      sometimes e need to compare samples taken from two population differeing in some characters. For example samples are drawn from two population namely \u201cnormal\u201d and liver disease patients\u2019; these to samples are then classified into two:<\/span><\/p>\r\n\r\n