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| The '''Wald test''' is a [[parametric statistics|parametric statistical test]] named after the [[Transylvania]]n statistician [[Abraham Wald]] with a great variety of uses. Whenever a relationship within or between data items can be expressed as a statistical model with parameters to be estimated from a sample, the Wald test can be used to test the true value of the parameter based on the sample estimate.
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| Suppose an economist, who has data on social class and shoe size, wonders whether social class is associated with shoe size. Say <math>\theta</math> is the average increase in shoe size for upper-class people compared to middle-class people: then the Wald test can be used to test whether <math>\theta</math> is 0 (in which case social class has no association with shoe size) or non-zero (shoe size varies between social classes). Here, <math>\theta</math>, the hypothetical difference in shoe sizes between upper and middle-class people in the whole population, is a parameter. An estimate of <math>\theta</math> might be the difference in shoe size between upper and middle-class people in the sample. In the Wald test, the economist uses the estimate and an estimate of variability (see below) to draw conclusions about the unobserved true <math>\theta</math>. Or, for a medical example, suppose smoking multiplies the risk of lung cancer by some number ''R'': then the Wald test can be used to test whether ''R'' = 1 (i.e. there is no effect of smoking) or is greater (or less) than 1 (i.e. smoking alters risk).
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| A Wald test can be used in a great variety of different models including models for [[dichotomous]] variables and models for [[Variable (mathematics)|continuous variable]]s.<ref>{{cite book |last=Harrell |first=Frank E., Jr. |year=2001 |title=Regression modeling strategies |publisher=Springer-Verlag |location=New York |isbn=0387952322 |chapter=Sections 9.2, 10.5 }}</ref>
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| == Mathematical details ==
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| Under the Wald statistical test, the [[maximum likelihood]] estimate <math>\hat\theta</math> of the parameter(s) of interest <math>\theta</math> is compared with the proposed value <math>\theta_0</math>, with the assumption that the difference between the two will be approximately [[normal distribution|normally distributed]]. Typically the square of the difference is compared to a [[chi-squared distribution]]. In the univariate case, the Wald [[statistic]] is
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| :<math>
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| \frac{ ( \widehat{ \theta}-\theta_0 )^2 }{\operatorname{var}(\hat \theta )}
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| </math>
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| which is compared against a [[chi-squared distribution]].
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| Alternatively, the difference can be compared to a [[normal distribution]]. In this case the test statistic is
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| :<math>\frac{\widehat{\theta}-\theta_0}{\operatorname{se}(\hat\theta)}</math>
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| where <math>\operatorname{se}(\widehat\theta)</math> is the [[Standard error (statistics)|standard error]] of the maximum likelihood estimate. A reasonable estimate of the standard error for the MLE can be given by <math> \frac{1}{\sqrt{I_n(MLE)}} </math>, where <math> I_n </math> is the [[Fisher information]] of the parameter.
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| In the multivariate case, a test about several parameters at once is carried out using a variance matrix.<ref>{{cite book |last=Harrell |first=Frank E., Jr. |year=2001 |title=Regression modeling strategies |publisher=Springer-Verlag |location=New York |isbn=0387952322 |chapter=Section 9.3.1 }}</ref> A common use for this is to carry out a Wald test on a [[categorical variable]] by recoding it as several dichotomous variables.
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| == Alternatives to the Wald test ==
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| The [[likelihood-ratio test]] can also be used to test whether an effect exists or not. The Wald test and the likelihood ratio test often give similar conclusions (as they are asymptotically equivalent), but they could disagree enough to lead to different conclusions.
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| There are several reasons to prefer the likelihood ratio test to the Wald test.<ref>{{cite book |last=Harrell |first=Frank E., Jr. |year=2001 |title=Regression modeling strategies |publisher=Springer-Verlag |location=New York |isbn=0387952322 |chapter=Section 9.3.3 }}</ref><ref>{{cite book |last=Collett |first=David |title=Modelling Survival Data in Medical Research |location=London |year=1994 |publisher=Chapman & Hall |isbn=0412448807 }}</ref><ref>{{cite book |last=Pawitan |first=Yudi |year=2001 |title=In All Likelihood |location=New York |publisher=Oxford University Press |isbn=0198507658 }}</ref> One is that the Wald test can give different answers to the same question, depending on how the question is phrased.<ref>{{cite journal |last=Fears |first=Thomas R. |last2=Benichou |first2=Jacques |last3=Gail |first3=Mitchell H. |year=1996 |title=A reminder of the fallibility of the Wald statistic |journal=[[The American Statistician]] |volume=50 |issue=3 |pages=226–227 |doi=10.1080/00031305.1996.10474384 }}</ref> For example, asking whether ''R'' = 1 is the same as asking whether log ''R'' = 0; but the Wald statistic for ''R'' = 1 is not the same as the Wald statistic for log ''R'' = 0 (because there is in general no neat relationship between the standard errors of ''R'' and log ''R''). Likelihood ratio tests will give exactly the same answer whether we work with ''R'', log ''R'' or any other [[monotonic]] transformation of ''R''. The other reason is that the Wald test uses two approximations (that we know the standard error, and that the distribution is [[Chi-squared distribution|chi-squared]]), whereas the likelihood ratio test uses one approximation (that the distribution is chi-squared).
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| Yet another alternative is the [[score test]], which has the advantage that it can be formulated in situations where the variability is difficult to estimate; e.g. the [[Cochran–Mantel–Haenzel test]] is a score test.<ref>{{cite book |last=Agresti |first=Alan |year=2002 |title=Categorical Data Analysis |publisher=Wiley |page=232 |isbn=0471360937 |edition=2nd }}</ref>
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| == Asymptotic properties ==
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| [[Robert F. Engle]] showed that the Wald test, the [[likelihood-ratio test]] and the [[Score test|Lagrange multiplier test]] (also known as the score test) are [[Asymptotic distribution|asymptotically equivalent]].<ref>{{cite book |title=Handbook of Econometrics |last=Engle |first=Robert F. |editor=Intriligator, M. D.; and Griliches, Z. |publisher=Elsevier |year=1983 |volume=II |pages=796–801 |chapter=Wald, Likelihood Ratio, and Lagrange Multiplier Tests in Econometrics |isbn=978-0-444-86185-6 }}</ref>
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| == See also ==
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| {{Colbegin}}
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| * [[Chow test]]
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| {{Colend}}
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| == References ==
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| <references/>
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| == External links ==
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| *[http://jeff560.tripod.com/w.html Wald test] on the [http://jeff560.tripod.com/mathword.html Earliest known uses of some of the words of mathematics]
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| {{statistics}}
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| {{DEFAULTSORT:Wald Test}}
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| [[Category:Statistical tests]]
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