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M obile C omputing G roup. A quick-and-dirty tutorial on the chi2 test for goodness-of-fit testing. Outline. The presentation follows the pyramid schema. Chi2 tests for GoF. Goodness-of-fit (GoF). Background -concepts. Background. Descriptive vs. inferential statistics

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M obile c omputing g roup l.jpg
Mobile Computing Group

A quick-and-dirty tutorial on the chi2 test for goodness-of-fit testing

Outline l.jpg

The presentation follows the pyramid schema

Chi2 tests for GoF

Goodness-of-fit (GoF)

Background -concepts

Background l.jpg

  • Descriptive vs. inferential statistics

    • Descriptive : data used only for descriptive purposes (use tables, graphs, measures of variability etc.)

    • Inferential : data used for drawing inferences, make predictions etc.

  • Sample vs. population

    • A sample is drawn from a population, assumed to have some characteristics.

    • The sample is often used to make inferences about the population (inferential statistics) :

      • Hypothesis testing

      • Estimation of population parameters

Background4 l.jpg

  • Statistic vs. parameter

    • A statistic is related (estimated from) a sample. It can be used for both descriptive and inferential purposes

    • A parameter refers to the whole population. A sample statistic is often used to infer a population parameter

      • Example : the sample mean may be used to infer the population mean (expected value)

  • Hypothesis testing

    • A procedure where sample data are used to evaluate a hypothesis regarding the population

    • A hypothesis may refer to several things : properties of a single population, relation between two populations etc.

    • Two statistical hypotheses are defined: a null H0 and an alternative H1

      • H0 is the often a statement of no effect or no difference. It is the hypothesis the researcher seeks to reject

Background5 l.jpg

  • Inferential statistical test

    • Hypothesis testing is carried out via an inferential statistic test :

      • Sample data are manipulated to yield a test statistic

      • The obtained value of the test statistic is evaluated with respect to a sampling distribution, i.e.,a theoretical probability distribution for the possible values of the test statistic

      • The theoretical values of the statistic are usually tabulated and let someone assess the statistical significance of the result of his statistical test

  • The goodness-of-fit is a type of hypothesis testing

    • devise inferential statistical tests, apply them to the sample, infer the matching of a theoretical distribution to the population distribution

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GoF as hypothesis testing

  • Hypothesis H0:

    • The sample is derived from a theoretical distribution F()

  • The sample data are manipulated to derive a test statistic

    • In the case of the chi2 statistic this includes aggregation of data into bins and some computations

  • The statistic, as computed from data, is checked against the sampling distribution

    • For the chi2 test, the sampling distribution is the chi2 distribution, hence the name

Goodness of fit l.jpg

  • Statistical tests and statistics : the big picture

EDF-based tests

Chi2 type tests

Specialized tests

e.g., KS test, Anderson-Darling test

e.g., Shapiro-Wilk test for normality

Generalized chi2 statistics

Classical chi2 statistics

Log-likelihood ratio statistic

Modified chi2 statistic

Pearson chi2 statistic

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Pearson chi2 statistic

  • M : number of bins

  • Oi (Ni):observed frequency in bin i

  • n : sample size

  • Ei (npi) : expected frequency in bin i according to the theoretical distribution F()

If X1, X2, X3…Xn , the random sample and F() the theoretical distribution under test,

the Pearson chi2 statistic is computed as:

Interpretation of chi2 statistic l.jpg
Interpretation of chi2 statistic

  • Theory says that the Pearson chi2 statistic follows a chi2 distribution, whose df are

    • M-1, when the parameters of the fitted distribution are given a priori (case 0 test)

    • Somewhere between M-1 and M-1-q, when the q parameters of the distribution are estimated by the sample data

    • Usually, the df for this case are taken to be M-1-q

  • Having estimated the value of the chi2 statistic X2 , I check the chi2 distribution with M-1 (M-1-q) df to find

    • What is the probability to get a value equal to or greater than the computed value X2, called p-value

    • If p > a, where a is the significance level of my test, the hypothesis is rejected, otherwise it is retained

    • Standard values for a are 0.1, 0.05, 0.01 – the higher a is the more conservative I am in rejecting the hypothesis H0

Example l.jpg

  • A die is rolled 120 times

  • 1 comes 20 times, 2 comes 14, 3 comes 18, 4 comes 17, 5 comes 22 and 6 comes 29 times

  • The question is: “Is the die biased?” –or better: “Do these data suggest that the die is biased?”

  • Hypothesis H0 : the die is not biased

    • Therefore, according to the null hypothesis these numbers should be distributed uniformly

    • F() : the discrete uniform distribution

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Example – cont.

  • Interpretation

    • The distribution of the test statistic has 5 df

    • The probability to get a value smaller or equal than 6.7 under a chi2 distribution with 5 df (p-value) is 0.75, which is < 1-a for all a in {0.01..0.1}.

    • Therefore the hypothesis that the die is not biased cannot be rejected

  • Computations:

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Interpretation of Pearson chi2

  • Graphical illustration

  • At 10% significance level, I would reject the hypothesis if the computed X2>9.24)

10% of the area under the curve






P-value :





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Properties of Pearson chi2 statistic

  • It can be estimated for both discrete and continuous variables

    • Holds for all chi2 statistics. Max flexibility but fails to make use of all available information for continuous variables

  • It is maybe the simplest one from computational point of view

  • As with all chi2 statistics, one needs to define number and borders of bins

    • These are generally a function of sample size and the theoretical distribution under test

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Bin selection

  • How many and which?

    • Different opinions in literature, no rigid proof of optimality

  • There seems to be convergence on the following aspects

    • Probability of bins

      • The bins should be chosen equiprobable with respect to the theoretical distribution under test

    • Minimum expected frequencies npi :

      • (Cramer, 46) : npi > 10, for all bins

      • (Cochran, 54) : npi > 1 for all bins, npi >= 5 for 80% of bins

      • (Roscoe and Byars,71)

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Bin selection

  • Relevance of bins M to sample size N

    • (Mann and Wald, 42), (Schorr, 74) : for large sample sizes

      1.88n2/5 < M < 3.76n2/5

    • (Koehler and Larntz,80) : for small sample size

      M>=3, n>=10 and n2/M>=10

    • (Roscoe and Byars, 71)

      • Equi-probable bins hypothesis : N > M when a = 0.01 and a = 0.05

      • Non-equiprobable bins : N>2M (a = 0.05) and N>4M (a=0.01)

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Bin selection

  • Bins vs. sample size according to Mann and Ward

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Bin selection : cont. vs. discrete






Equi-probable bins easy to select






Bin i


Less straightforward to define equi-probable bins








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  • D.J. Sheskin, Handbook of parametric and nonparametric statistical procedures

    • Introduction (descriptive vs. inferential statistics, hypothesis testing, concepts and terminology)

    • Test 8 (chap. 8) – The Chi-Square Goodness-of-Fit Test (high-level description with examples and discussion on several aspects)

  • R. Agostino, M. Stephens, Goodness-of-fit techniques

    • Chapter 3 – Tests of Chi-square type

      • Reviews the theoretical background and looks more generally at chi2 tests, not only the Pearson test.

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  • S. Horn, Goodness-of-Fit tests for discrete data: A review and an Application to a Health Impairment scale

    • Good discussion of the properties and pros/cons of most goodness-of-fit tests for discrete data

    • accessible, tutorial-like

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