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Goodness of Fit Tests. The goal of χ 2 goodness of fit tests is to test is the data comes from a certain distribution. There are various situations to which these tests apply. The first situation we will explore is when we observe count data in k different categories.

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Goodness of fit tests
Goodness of Fit Tests

  • The goal of χ2 goodness of fit tests is to test is the data comes from a

    certain distribution.

  • There are various situations to which these tests apply.

  • The first situation we will explore is when we observe count data in

    k different categories.

  • The aim is to test the null hypothesis that the probabilities of the k

    categories are p1, p2,…,pk.

  • We distinguish between two cases.

STA261 week 12


Chi squared test case 1
Chi-Squared Test - Case 1

  • The null hypothesis completely specifies the probabilities of each of

    the k categories.

  • For each category we calculate the expected count Ei = npi.

  • The test statistic and its distribution are…

STA261 week 12


Example
Example

  • The statistic department at U of T offers introductory courses for

    students from other disciplines. The department believes that 40% of

    the students are math major, 30% are computer science, 20%

    biology and 10% chemistry. A random sample of 120 students

    revealed 52, 38, 21, and 9 from the four majors above. Does this

    data support the department claim?

STA261 week 12


Chi squared test case 2
Chi-Squared Test - Case 2

  • The null hypothesis does not fully specify the probabilities.

  • In this case the probabilities of the different categories may be functions of other parameters.

  • First use the sample data to estimate r unknown parameters.

  • Then use the estimated parameters to estimate the k probabilities.

  • For each category, calculate the estimated expected count.

  • The test statistic is…

STA261 week 12


Example1
Example

  • A farmer believes that the number of eggs a chicken will give per

    day has a Poisson(λ) distribution. He observed the following data….

STA261 week 12


Remark
Remark

  • In many cases we will observe data that are not categorized and we

    would want to test if the data comes from a certain distribution.

  • If the distribution we are testing is discrete the values of the variable

    will be the actual categories.

  • However, if the variable takes infinite possible values, the grouping

    should be done so that the expected frequency in each category is at

    least 5.

  • If the distribution we are testing is continuous we need to group the

    measurement of the random variable of interest into k intervals.

    Very often the choice of cells is done arbitrarily.

  • χ2 tests has low power when they are applied to continuous data, in which case we can use other tests.

STA261 week 12


Example2
Example

STA261 week 12


Kolmogorov smirnov goodness of fit test
Kolmogorov-Smirnov Goodness-of-Fit Test

  • K-S test is also called the Kolmogorov-Smirnov D test.

  • The K-S goodness-of-fit test tests whether or not a given

    distribution is not significantly different from one hypothesized.

  • It is a more powerful alternative to chi-square goodness-of-fit tests.

  • The test statistic in the K-S test is based on the largest absolute

    difference between the cumulative observed proportion and the

    cumulative proportion expected on the basis of the hypothesized

    distribution.

STA261 week 12


Contingency tables
Contingency Tables

  • The goal is to test if two categorical variables are independent.

  • The row variable has r categories while the column variable has c categories.

  • The data is the count of observations in the rxc table…

  • The null hypothesis states that the row variable and the column

    variable are independent. The alternative states that the variables are

    dependent.

  • To conduct the test, we calculate the expected count for each cell…

  • The test statistic and its distribution is….

STA261 week 12


Example3
Example

STA261 week 12


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