More Tables of the Incomplete Gamma-Function Ratio and of Percentage Points of the Chi-Square Distribution

Harter, H. Leon

doi:10.21236/ad0607403

Cited by 77 publications

(35 citation statements)

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“…For example, there are many contributions that provide tables of quantiles for various levels of significance and degrees of freedom at various levels of accuracy. Some of the earlier contributions include those from Pearson (1922), Fisher (1928, Thompson (1941), Merrington (1941), Aroian (1943), Goldberg and Levine (1946), Hald and Sinkbaek (1950), Vanderbeck and Cooke (1961), Harter (1964) and Krauth and Steinebach (1976). While these contributions give tables of quantiles, others provide simple formulae for calculating these values for any level of significance; see, for example, Wilson and Hilferty (1931) and Heyworth (1976).…”

Section: Overview Of Pearson's Chi-squared Statistic and Its P-valuementioning

confidence: 99%

Exploring How to Simply Approximate the P-value of a Chi-squared Statistic

Beh

2018

AJS

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Calculating the p-value of any test statistic is of paramount importance to all statistically minded researchers across all areas of study. Many, these days, take for granted how the p-value is calculated and yet it is a pivotal quantity in all forms of statistical analysis. For the study of 2×2 tables where dichotomous variables are assessed for association, the chi-squared statistic, and its p-value, are fundamental quantities to all analysts, especially those in the health and allied disciplines. Examining the association between dichotomous variables is easily achieved through a very simple formula for the chi-squared statistic and yet the p-value of this statistic requires far more computational effort. This paper proposes and explores a very simple approximation of the p-value for a chi-squared statistic given its degrees of freedom. After providing a review of a variety of common ways for determining the quantile of the chi-squared distribution given the level of significance and degrees of freedom, we shall derive an approximation based on the classic quantile formula given in 1977 by D. C. Hoaglin. We examine this approximation using a simple 2×2 contingency table example then show that it is extremely precise for all chi-squared values ranging from 0 to 50.

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Section: Overview Of Pearson's Chi-squared Statistic and Its P-valuementioning

confidence: 99%

Exploring How to Simply Approximate the P-value of a Chi-squared Statistic

Beh

2018

AJS

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“…Numerical confidence bounds may be obtained with the aid of a t'able of percentage points of the chi-square distribution [see, for example, Harter (1964)]. …”

Section: Confidekce Bounds Forthe Scale Parametermentioning

confidence: 99%

Conditional Maximum-Likelihood Estimators, from Singly Censored Samples, of the Scale Parameters of Type II Extreme-Value Distributions

Harter

Moore

1968

Technometrics

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Use of the functional relationships between the exponential and the Type II asymptotic distributions of largest and smallest values enables one to obtain conditional maximum-likelihood estimators, from singly censored samples, of the scale parameters (characteristic largest and characteristic smallest values) of the Type II asymptotic distributions of largest and smallest values, Fl(y; v,, , K) = exp [-(y/v,)-"] and F&c; Q , K) = 1 -exp [ -(z/v~)-"1, by simple transformations of the corresponding estimator, a,,,, , of the scale parameter of the exponential distribution, based on the first m order statistics of a sample of size n. Use is made of the fact that 0, ( K = ii:'" and v^, [ I< = -&,A'k, where 2m&Jt? has the chi-square distribution with 2m degrees of freedom, to set confidence bounds on the scale parameters, v,, and 01 , of the Type II asymptotic distributions of largest and smallest values.The probability densities of 0, ( K and & 1 K, each of which for given m is the same for any n > m, are obtained by simple transformations of that of i,,. The expected values of &,I K and G1 1 K are determined, and from them the unbiasing factors by which 0, 1 K and 0, 1 K must be multiplied to obtain unbiased estimators, 5" 1 K and 51 1 I<. Expressions are found for the variances of these unbiased estimators and for the Cramer-Rao lower bounds. Values of the unbiasing factors, the variances of the unbiased estimators, and their efficiency relative to the Cram&-Rao lower bound, all of which are independent of n, are tabled for m = 1(1)20(4)100 and K = 0.5(0.5)4.0, 5.0. A numerical example and some remarks on applications are included.

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“…Tables of c were constructed for the same set of values used in Section 2, again using the Harter [5] and Dixon and Massey [2] tables. From Figure 2 we see that n = 70 is required to guarantee with probability .90 that the usual confidence interval with confidence coefficient .99 for the standard deviation will be shorter than .5u.…”

Section: (32)mentioning

confidence: 99%

Some Graphs Useful for Statistical Inference

Guenther

Thomas

1965

Journal of the American Statistical Association

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ANDThe determination of sample size to meet certain probability requirements is a problem which often faces an experimenter when obtaining a confidence interval or testing a hypothesis. This paper includes some new graphs useful in selecting sample size and gives a reference to a new textbook which includes a number of other similar type graphs. The method of construction is explained in detail and examples are included.

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More Tables of the Incomplete Gamma-Function Ratio and of Percentage Points of the Chi-Square Distribution

Cited by 77 publications

References 1 publication

Exploring How to Simply Approximate the P-value of a Chi-squared Statistic

Exploring How to Simply Approximate the P-value of a Chi-squared Statistic

Conditional Maximum-Likelihood Estimators, from Singly Censored Samples, of the Scale Parameters of Type II Extreme-Value Distributions

Some Graphs Useful for Statistical Inference

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