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Chi-Square Significance Tests

Chi-Square Significance Tests

Key Concepts and Terms

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  • Types of Chi-Square
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    • Pearson's chi-square is by far the most common type ofchi-square significance test. If simply "chi-square" is mentioned, itis probably Pearson's chi-square. This statistic is used to test thehypothesis of no association of columns and rows in tabular data. Itcan be used even with nominal data. Note that chi square is more likelyto establish significance to the extent that (1) the relationship isstrong, (2) the sample size is large, and/or (3) the number of valuesof the two associated variables is large. A chi-square probability of.05 or less is commonly interpreted by social scientists asjustification for rejecting the null hypothesis that the row variableis unrelated (that is, only randomly related) to the column variable.Its calculation, which is the sum of observed minus expected countsquared and divided by the expected, is illustrated in the figure belowand is discussed in detail in the FAQ section.

        / g4 b0 t. ~/ {# d9 ]www.spsschina.com ,spss论坛,数据分析,数据调查,市场调研,SPSS,PLS,DEA,spss下载,spss 下载,spss 教程,spss软件,spss中文版下载,spss免费下载,spss是什么,spss13.0下载,SPSS教程,Spss视频,数据分析师,数据分析论坛,数据分析软件For this example, the SPSS output from the Analyze, Descriptive Statistics, Crosstabs menu choice looks like this:spss论坛|spss下载|spss视频|Amos|SEM|SAS|Matlab|Eviews& u$ \" L( W6 X& B' E5 F5 s
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      • Yates' correction is an arbitrary, conservativeadjustment to chi-square when applied to tables with one or more cellswith frequencies less than five. It is only applied to 2 by 2 tables.Some authors also apply it to all 2 by 2 tables since the correctiongives a better approximation to the binomial distribution. Yates'correction is conservative in the sense of making it more difficult toestablish significance. SPSS. Some computer packages label Yates'correction as continuity corrected chi-square in their output. Its calculation is also discussed below.

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    • Chi-square goodness-of-fit test. Thegoodness-of-fit test is simply a different use of Pearsonianchi-square. It is used to test if an observed distribution conforms toany other distribution, such as one based on theory (ex., if theobserved distribution is not significantly different from a normaldistribution) or one based on some other known distribution (ex., ifthe observed distribution is not significantly different from a knownnational distribution based on Census data). The Kolmogorov-Smirnov goodness-of-fit test is preferred for interval data, for which it is more powerful than chi-square goodness-of-fit.An example of using the chi-square goodness-of-fit test to test if asample distribution of response is different from the distributionexpected on the basis of the population is located here,implemented as an Excel spreadsheet. Specifically, the example tests ifthe distribution of sample survey returns from field offices by regionis not significantly different from what would be expected given theknown (population) number of actual field offices by region.
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    • Likelihood ratio chi-square test, also called thelikelihood test or G test, is an alternative procedure to test thehypothesis of no association of columns and rows in nominal-leveltabular data. It is supported by SPSS output and is based on maximumlikelihood estimation. Though computed differently, likelihood ratiochi-square is interpreted the same way. For large samples, likelihoodratio chi-square will be close in results to Pearson chi-square. Evenfor smaller samples, it rarely leads to different substantive results.SPSS will print likelihood ratio chi-square in the "Chi-Square Tests"table of output from the Analyze, Descriptives, Crosstabs menuselection.www.spsschina.com ,spss论坛,数据分析,数据调查,市场调研,SPSS,PLS,DEA,spss下载,spss 下载,spss 教程,spss软件,spss中文版下载,spss免费下载,spss是什么,spss13.0下载,SPSS教程,Spss视频,数据分析师,数据分析论坛,数据分析软件. x/ T5 m# h7 e
      For the example above, which has 1 degree of freedom, the computedlikelihood ratio value of 3.3980 is significant at the .065 level. Thiscompares to the .068 level for Pearson chi-square for the same table.Continuity-corrected chi-square is .144 for the table. (All for 2-sidedtests). + j9 r( C' |. n3 P% i( e: ^
      www.spsschina.com ,spss论坛,数据分析,数据调查,市场调研,SPSS,PLS,DEA,spss下载,spss 下载,spss 教程,spss软件,spss中文版下载,spss免费下载,spss是什么,spss13.0下载,SPSS教程,Spss视频,数据分析师,数据分析论坛,数据分析软件$ O( m8 @3 V1 V( z) B' L) o* X
    • Mantel-Haenszel chi-square, also called the Mantel-Haenszel test for linear association or linear by linear association chi-square,unlike ordinary and likelihood ratio chi-square, is an ordinal measureof significance. It is preferred when testing the significance oflinear relationship between two ordinal variables because it is morepowerful than Pearson chi-square (more likely to establish linearassociation). Mantel-Haenzel chi-square is not appropriate for nominalvariables. If found significant, the interpretation is that increasesin one variable are associated with increases (or decreases fornegative relationships) in the other greater than would be expected bychance of random sampling. Like other chi-square statistics, M-Hchi-square should not be used with tables with small cell counts. spss论坛|spss下载|spss视频|Amos|SEM|SAS|Matlab|Eviews& f) b6 m4 |1 f$ ]. R
      • Stratified analysis, also called blocked analysis andmatched analysis, is a form of control variable analysis conducted withthe Mantel-Haenszel coefficient. For each of k categories of a controlvariable (called the stratification variable), a 2-by-2 tableis created for the independent and dependent variables. Thestratification variable need not be ordinal but it is assumed that therow and column marginals be the same for each of the k 2-by-2 tables, acircumstance which occurs mainly in experimental situations. TheMantel-Haenszel chi-square coefficient tests whether the common odds ratioacrossthe k strata is 1.0, indicating no effect of the stratificationvariable. SPSS provides a macro (mh.sps) for Mantel-Haenszel stratifiedanalysis which outputs M-H chi-square and its significance..

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    • SPSS Output. To obtain chi-square in SPSS: SelectAnalyze, Descriptive Statistics, Crosstabs; select row and columnvariables; slick Statistics; select Chi-square.
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      • Chi-square with control variables: In the context ofcrosstabulation, use of a control variable creates one subtable(similar to the overall table) for each value of the control variable.Evaluation of the subtable with chi-square is identical to evaluationof the main table. There is a control effect if at least one subtableis non-significant. In SPSS, move the control variable to the Layer 1box when selecting variables. Use of multiple control variables ispossible.

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    • Type of significance estimate. The Exact button inthe SPSS dialog above allows the researcher to select among asymptotic,exact, or Monte-Carlo estimates of the significance of theKolmogorov-Smirnov test value. These three types of estimates arediscussed separately in the section on significance testing. This requires that the SPSS Exact Tests add-on module be installed.
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3 u( i- j( [/ n2 A* {8 ?: o1 Y3 g; Kspsschina.cnAssumptions
  • Random sample data are assumed. As with all significancetests, if you have population data, then any table differences are realand therefore significant. If you have non-random sample data,significance cannot be established, though significance tests arenonetheless sometimes utilized as crude "rules of thumb" anyway.& x' c# O0 a  r$ l0 L
  • A sufficiently large sample size is assumed, as inall significance tests. Applying chi-square to small samples exposesthe researcher to an unacceptable rate of Type II errors. There is noaccepted cutoff. Some set the minimum sample size at 50, while otherswould allow as few as 20. Note chi-square must be calculated on actualcount data, not substituting percentages, which would have the effectof pretending the sample size is 100.SPSS,spss下载,spss 下载,spss 教程,spss软件,spss中文版下载,spss免费下载,数据分析师,数据分析论坛,数据分析软件,spss13.0下载,SPSS教程,Spss视频amos ,sem analysis,spss function,spss net,spss software,spss statistical,数据分析师,数据分析论坛,数据分析软件,PLS,DEA,0 a( O- V$ E3 u9 D! i1 m: c/ M
  • Adequate cell sizes are also assumed. Some require5 or more, some require more than 5, and others require 10 or more. Acommon rule is 5 or more in all cells of a 2-by-2 table, and 5 or morein 80% of cells in larger tables, but no cells with zero count. Whenthis assumption is not met, Yates' correction is applied.
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  • Independence. Observations must be independent.The same observation can only appear in one cell. This means chi-squarecannot be used to test correlated data (ex., before-after, matchedpairs, panel data).spss论坛|spss下载|spss视频|Amos|SEM|SAS|Matlab|Eviews+ P7 |3 a, a, Z' p4 C  W+ R
  • Similar distribution. Observations must have the same underlying distribution.spsschina.cn: \% y" w$ {# g
  • Known distribution. The hypothesized distributionis specified in advance, so that the number of observations that areexpected to appear each cell in the table can be calculated withoutreference to the observed values. Normally this expected value is thecrossproduct of the row and column marginals divided by the samplesize.www.spsschina.com ,spss论坛,数据分析,数据调查,市场调研,SPSS,PLS,DEA,spss下载,spss 下载,spss 教程,spss软件,spss中文版下载,spss免费下载,spss是什么,spss13.0下载,SPSS教程,Spss视频,数据分析师,数据分析论坛,数据分析软件( Q  d+ T- w6 _! w( l7 J+ Z
  • Non-directional hypotheses are assumed. Chi-squaretests the hypothesis that two variables are related only by chance. Ifa significant relationship is found, this is not equivalent toestablishing the researcher's hypothesis that A causes B, or that Bcauses A.
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  • Finite values. Observations must be grouped in categories.spss论坛|spss下载|spss视频|Amos|SEM|SAS|Matlab|Eviews8 A# K% A  R$ _
  • Normal distribution of deviations (observed minus expected values) is assumed. Note chi-square is a nonparametric test in the sense that is does not assume the parameter of normal distribution for the data -- only for the deviations.% P6 }$ Y: s, @9 a( D
  • Data level. No assumption is made about level of data. Nominal, ordinal, or interval data may be used with chi-square tests.

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Frequently Asked Questions
  • How is Pearsonian chi-square calculated for tabular data?
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      Calculation of this form of chi-square requires four steps:spss论坛|spss下载|spss视频|Amos|SEM|SAS|Matlab|Eviews0 o  \+ v5 T) E: K; F% q
      • Computing the expected frequencies. For each cell in thetable, the expected frequency must be calculated. The expectedfrequency for a given column is the column total divided by n, thesample size. The expected frequency for a given row is the row totaldivided by n. The expected frequency for a cell is the columnexpectation times the row expectation times n. This formula reduces tothe expected frequency for a given cell equaling its row total timesits column total, divided by n.
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      • Application of the chi-square formula. Let O bethe observed value of each cell in a table. Let E be the expected valuecalculated in the previous step. For each cell, subtract E from O, thensquare the result and then divide by E. Do this for every cell and sumall the results. This is the chi-square value for the table. If Yates'correction for continuity is to be applied, due to cell counts below 5,the calculation is the same except for each cell, subtract anadditional .5 from the difference of O - E, prior to squaring and thendividing by E. This reduces the size of the calculated chi-squarevalue, making a finding of significance less likely -- a penalty deemedappropriate for tables with low counts in some cells.www.spsschina.com ,spss论坛,数据分析,数据调查,市场调研,SPSS,PLS,DEA,spss下载,spss 下载,spss 教程,spss软件,spss中文版下载,spss免费下载,spss是什么,spss13.0下载,SPSS教程,Spss视频,数据分析师,数据分析论坛,数据分析软件: |  N& h  l) J; X
      • Calculate the degrees of freedom. The chi-square value is not interpretable directly but must be compared to a table of the chi-square distribution.The columns of this table are alternative significance levels (.001,.01, .05, etc.) and the rows are degrees of freedom (df). For a table,df = (r - 1)*(c - 1) , where r is the number of rows and c is thenumber of columns. That is, if you know the column and row totals, whenall cells are filled in except one row and one column, these may becalculated from the information already given.www.spsschina.com ,spss论坛,数据分析,数据调查,市场调研,SPSS,PLS,DEA,spss下载,spss 下载,spss 教程,spss软件,spss中文版下载,spss免费下载,spss是什么,spss13.0下载,SPSS教程,Spss视频,数据分析师,数据分析论坛,数据分析软件" `9 j. n% ?4 _( R6 F' \# s
      • Using the chi-square table. A chi-square table, which in effect is built into statistical software packages, gives a critical value.The calculated chi-square value must be greater than the critical valueto reject the null hypothesis that the row variable is unrelated to thecolumn variable, at the level of significance selected by reading downthe appropriate column in the chi-square table (ex., the .05significance column). In practice, computer programs are used in placeof chi-square tables, and computer printout shows the significancelevel (often labeled p) directly, but the interpretation is the same.
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  • My statistics program prints out the chi-squarecontribution of each cell in the table. Can this be used to establishthe significance of each cell?
      No, a different procedure is needed for that purpose. However, cellcontributions are useful in determining which ranges of the twovariables depicted in a table are contributing the most to the overallrelationship.

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  • One of my variables is a multi-response item. What do I do?
      One method is to compute a new nominal variable in which everycombination of responses is a separate value (use the IF statements inSPSS), then you may use chi-square or nominal measures of association.Alternatively, you might consider using the SPSS CATEGORIES moduleafter recoding your multi-response item as a set of separate variables.CATEGORIES then gives you the canonical correlation coefficient between your multiple response set as independents and another variable such as income level as dependent.

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. o( _) a1 L" v" j- H1 Kwww.spsschina.com ,spss论坛,数据分析,数据调查,市场调研,SPSS,PLS,DEA,spss下载,spss 下载,spss 教程,spss软件,spss中文版下载,spss免费下载,spss是什么,spss13.0下载,SPSS教程,Spss视频,数据分析师,数据分析论坛,数据分析软件Bibliography
  • Agresti, Alan (1996). Introduction to categorical data analysis. NY: John Wiley and Sons. Agresti discusses Mantel-Haenszel chi-square stratified analysis on pp. 231-236.
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  • Levin, Irwin P. (1999). Relating statistics and experimental design.Thousand Oaks, CA: Sage Publications. Quantitative Applications in theSocial Sciences series #125. Elementary introduction covers t-tests andvarious simple ANOVA designs. Some additional discussion of chi-square,significance tests for correlation and regression. and non-parametrictests such as the runs test, median test, and Mann-Whitney U test., @" U5 [* q* Z5 j  E: D2 C2 }
  • Lieberman, Bernhardt, ed. (1971). Contemporary problems in statistics. NY: Oxford. Section 5 deals with assumptions of chi-square procedures.

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