On the use of chi-square analyses in studies of resource utilization

1991 ◽  
Vol 21 (1) ◽  
pp. 58-65 ◽  
Author(s):  
Dennis E. Jelinski
Keyword(s):  
Resource Utilization ◽  
Goodness Of Fit ◽  
Small Sample ◽  
Homogeneity Test ◽  
Type I ◽  
Sample Sizes ◽  
Goodness Of Fit Test ◽  
Chi Square ◽  
Test Of Homogeneity ◽  
Small Sample Sizes

Chi-square (χ2) tests are analytic procedures that are often used to test the hypothesis that animals use a particular food item or habitat in proportion to its availability. Unfortunately, several sources of error are common to the use of χ2 analysis in studies of resource utilization. Both the goodness-of-fit and homogeneity tests have been incorrectly used interchangeably when resource availabilities are estimated or known apriori. An empirical comparison of the two methods demonstrates that the χ2 test of homogeneity may generate results contrary to the χ2 goodness-of-fit test. Failure to recognize the conservative nature of the χ2 homogeneity test, when "expected" values are known apriori, may lead to erroneous conclusions owing to the increased possibility of committing a type II error. Conversely, proper use of the goodness-of-fit method is predicated on the availability of accurate maps of resource abundance, or on estimates of resource availability based on very large sample sizes. Where resource availabilities have been estimated from small sample sizes, the use of the χ2 goodness-of-fit test may lead to type I errors beyond the nominal level of α. Both tests require adherence to specific critical assumptions that often have been violated, and accordingly, these assumptions are reviewed here. Alternatives to the Pearson χ2 statistic are also discussed.

scholarly journals Comparative evaluation of goodness of fit tests for normal distribution using simulation and empirical data

2020 ◽  
Vol 57 (2) ◽  
pp. 237-251
Author(s):  
Achilleas Anastasiou ◽  
Alex Karagrigoriou ◽  
Anastasios Katsileros
Keyword(s):  
Sample Size ◽  
Normal Distribution ◽  
Empirical Data ◽  
Goodness Of Fit ◽  
Small Sample Size ◽  
Small Sample ◽  
Type I ◽  
Sample Sizes ◽  
Omnibus Test ◽  
Wide Range

SummaryThe normal distribution is considered to be one of the most important distributions, with numerous applications in various fields, including the field of agricultural sciences. The purpose of this study is to evaluate the most popular normality tests, comparing the performance in terms of the size (type I error) and the power against a large spectrum of distributions with simulations for various sample sizes and significance levels, as well as through empirical data from agricultural experiments. The simulation results show that the power of all normality tests is low for small sample size, but as the sample size increases, the power increases as well. Also, the results show that the Shapiro–Wilk test is powerful over a wide range of alternative distributions and sample sizes and especially in asymmetric distributions. Moreover the D’Agostino–Pearson Omnibus test is powerful for small sample sizes against symmetric alternative distributions, while the same is true for the Kurtosis test for moderate and large sample sizes.

scholarly journals Chi-square Reference Meanings: a Historical-epistemological Overview

Revemop ◽  
2021 ◽  
Vol 3 ◽  
pp. e202108
Author(s):  
Jesus Guadalupe Lugo-Armenta ◽  
Luis Roberto Pino-Fan ◽  
Blanca Rosa Ruiz Hernandez
Keyword(s):  
Present Article ◽  
Goodness Of Fit ◽  
The Other ◽  
Inferential Reasoning ◽  
Goodness Of Fit Test ◽  
Chi Square ◽  
Test Of Homogeneity ◽  
Semiotic Approach ◽  
The One ◽  
Epistemic Criteria

The present article shows a historical-epistemological study on the Chi-square statistic. In which theoretical-methodological notions from the Onto-Semiotic Approach (OSA) of mathematical cognition and instruction were used to identify four problems that have been key to the evolution of the Chi-square statistic: the Goodness-of-fit-test, the test of independence, the test of homogeneity and distribution. Furthermore, various meanings of the Chi-square statistic were recognized in the mathematical-statistical practices that are used to solve each of those problems. These meanings could help to establish epistemic criteria that allow, on the one hand, to propose progressive levels of inferential reasoning for the statistic (from informal to formal); and on the other hand, to design tasks oriented to promote the understanding of the diverse meanings of the Chi-square.

Alternatives to the Chi-Square Test for Evaluating Rank Histograms from Ensemble Forecasts

2005 ◽  
Vol 20 (5) ◽  
pp. 789-795 ◽  
Author(s):  
Kimberly L. Elmore
Keyword(s):  
Goodness Of Fit ◽  
Small Sample ◽  
Chi Square ◽  
Ensemble Forecasts ◽  
Statistical Fluctuations ◽  
Rank Histogram ◽  
Anderson Darling ◽  
Discrete Uniform ◽  
Small Sample Sizes ◽  
Von Mises

Abstract Rank histograms are a commonly used tool for evaluating an ensemble forecasting system’s performance. Because the sample size is finite, the rank histogram is subject to statistical fluctuations, so a goodness-of-fit (GOF) test is employed to determine if the rank histogram is uniform to within some statistical certainty. Most often, the χ2 test is used to test whether the rank histogram is indistinguishable from a discrete uniform distribution. However, the χ2 test is insensitive to order and so suffers from troubling deficiencies that may render it unsuitable for rank histogram evaluation. As shown by examples in this paper, more powerful tests, suitable for small sample sizes, and very sensitive to the particular deficiencies that appear in rank histograms are available from the order-dependent Cramér–von Mises family of statistics, in particular, the Watson and Anderson–Darling statistics.

A Monte Carlo Comparison of Seven ε-Adjustment Procedures in Repeated Measures Designs With Small Sample Sizes

1994 ◽  
Vol 19 (1) ◽  
pp. 57-71 ◽  
Author(s):  
Stephen M. Quintana ◽  
Scott E. Maxwell
Keyword(s):  
Repeated Measures ◽  
Type I Error ◽  
Error Rates ◽  
Small Sample ◽  
Small Samples ◽  
Type I ◽  
Sample Sizes ◽  
Type I Error Rates ◽  
Repeated Measures Designs ◽  
Small Sample Sizes

The purpose of this study was to evaluate seven univariate procedures for testing omnibus null hypotheses for data gathered from repeated measures designs. Five alternate approaches are compared to the two more traditional adjustment procedures (Geisser and Greenhouse’s ε̂ and Huynh and Feldt’s ε̃), neither of which may be entirely adequate when sample sizes are small and the number of levels of the repeated factors is large. Empirical Type I error rates and power levels were obtained by simulation for conditions where small samples occur in combination with many levels of the repeated factor. Results suggested that alternate univariate approaches were improvements to the traditional approaches. One alternate approach in particular was found to be most effective in controlling Type I error rates without unduly sacrificing power.

Comparing Alternative Corrections for Bias in the Bias-Corrected Bootstrap Test of Mediation

2021 ◽  
pp. 016327872110243
Author(s):  
Donna Chen ◽  
Matthew S. Fritz
Keyword(s):  
Effect Size ◽  
Statistical Power ◽  
Type I Error ◽  
Small Sample ◽  
Medium Effect ◽  
Type I ◽  
Sample Sizes ◽  
Bootstrap Test ◽  
Small Effect Size ◽  
Small Sample Sizes

Although the bias-corrected (BC) bootstrap is an often-recommended method for testing mediation due to its higher statistical power relative to other tests, it has also been found to have elevated Type I error rates with small sample sizes. Under limitations for participant recruitment, obtaining a larger sample size is not always feasible. Thus, this study examines whether using alternative corrections for bias in the BC bootstrap test of mediation for small sample sizes can achieve equal levels of statistical power without the associated increase in Type I error. A simulation study was conducted to compare Efron and Tibshirani’s original correction for bias, z 0, to six alternative corrections for bias: (a) mean, (b–e) Winsorized mean with 10%, 20%, 30%, and 40% trimming in each tail, and (f) medcouple (robust skewness measure). Most variation in Type I error (given a medium effect size of one regression slope and zero for the other slope) and power (small effect size in both regression slopes) was found with small sample sizes. Recommendations for applied researchers are made based on the results. An empirical example using data from the ATLAS drug prevention intervention study is presented to illustrate these results. Limitations and future directions are discussed.

scholarly journals Likelihood ratio test between two groups of castor oil plant traits

2016 ◽  
Vol 46 (7) ◽  
pp. 1158-1164
Author(s):  
Betania Brum ◽  
Sidinei José Lopes ◽  
Daniel Furtado Ferreira ◽  
Lindolfo Storck ◽  
Alberto Cargnelutti Filho
Keyword(s):  
Likelihood Ratio ◽  
Likelihood Ratio Test ◽  
Type I Error ◽  
Plant Traits ◽  
Small Sample ◽  
Ratio Test ◽  
Type I ◽  
Sample Sizes ◽  
Oil Plant ◽  
Small Sample Sizes

ABSTRACT: The likelihood ratio test (LRT), to the independence between two sets of variables, allows to identify whether there is a dependency relationship between them. The aim of this study was to calculate the type I error and power of the LRT for determining independence between two sets of variables under multivariate normal distributions in scenarios consisting of combinations of 16 sample sizes; 40 combinations of the number of variables of the two groups; and nine degrees of correlation between the variables (for the power). The rate of type I error and power were calculate at 640 and 5,760 scenarios, respectively. A performance evaluation of the LRT was conducted by computer simulation by the Monte Carlo method, using 2,000 simulations in each scenario. When the number of variables was large (24), the TRV controlled the rate of type I errors and showed high power in sizes greater than 100 samples. For small sample sizes (25, 30 and 50), the test showed good performance because the number of variables did not exceed 12.

scholarly journals Permutation-based methods for mediation analysis in studies with small sample sizes

PeerJ ◽  
2020 ◽  
Vol 8 ◽  
pp. e8246
Author(s):  
Miranda E. Kroehl ◽  
Sharon Lutz ◽  
Brandie D. Wagner
Keyword(s):  
Indirect Effect ◽  
Mediation Analysis ◽  
Type I Error ◽  
Permutation Test ◽  
Error Rates ◽  
Small Sample ◽  
Type I ◽  
Permutation Testing ◽  
Sample Sizes ◽  
Small Sample Sizes

Background Mediation analysis can be used to evaluate the effect of an exposure on an outcome acting through an intermediate variable or mediator. For studies with small sample sizes, permutation testing may be useful in evaluating the indirect effect (i.e., the effect of exposure on the outcome through the mediator) while maintaining the appropriate type I error rate. For mediation analysis in studies with small sample sizes, existing permutation testing methods permute the residuals under the full or alternative model, but have not been evaluated under situations where covariates are included. In this article, we consider and evaluate two additional permutation approaches for testing the indirect effect in mediation analysis based on permutating the residuals under the reduced or null model which allows for the inclusion of covariates. Methods Simulation studies were used to empirically evaluate the behavior of these two additional approaches: (1) the permutation test of the Indirect Effect under Reduced Models (IERM) and (2) the Permutation Supremum test under Reduced Models (PSRM). The performance of these methods was compared to the standard permutation approach for mediation analysis, the permutation test of the Indirect Effect under Full Models (IEFM). We evaluated the type 1 error rates and power of these methods in the presence of covariates since mediation analysis assumes no unmeasured confounders of the exposure–mediator–outcome relationships. Results The proposed PSRM approach maintained type I error rates below nominal levels under all conditions, while the proposed IERM approach exhibited grossly inflated type I rates in many conditions and the standard IEFM exhibited inflated type I error rates under a small number of conditions. Power did not differ substantially between the proposed PSRM approach and the standard IEFM approach. Conclusions The proposed PSRM approach is recommended over the existing IEFM approach for mediation analysis in studies with small sample sizes.

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