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I understand that the output of, for example, summary(lm(Fertility ~ . , data = swiss)) is a table of $\hat\beta$s, standard errors, t-statistics, and p-values for two-sided tests of the null hypothesis that each parameter is equal to 0.

But, what should I actually call this method of evaluating a linear model when presenting it to non-statisticians? If I say I used a t-test, it will confuse the hell out of them.

We have the term ANOVA to describe testing hypotheses about coefficients by using F-statistics when we don't care about the direction of change. Is there a corresponding concise, unambiguous term for testing hypotheses about linear coefficients by using their t-statistics when we do care about the direction of change?

Thanks.

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    $\begingroup$ In addition to gung's and Glen's excellent answers below, you might find this post informative. $\endgroup$ – COOLSerdash Jun 25 '13 at 6:32
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    $\begingroup$ @COOLSerdash Good answers by gung and Glen_b, indeed, but it would have been nice if the question relating to presenting the results to non-statisticians was addressed. As the OP says, "If I say I used a t-test, it will confuse the hell out of them". If I have understood the situation correctly, the question still remains as to how the OP is to circumvent this communication problem. $\endgroup$ – Graeme Walsh Jun 25 '13 at 9:26
  • $\begingroup$ (cont...) Assuming that the communication tag is appropriate for this question then a related CV question is this one. Although related, however, it doesn't exactly get to the heart of what's being asked here. $\endgroup$ – Graeme Walsh Jun 25 '13 at 9:47
  • $\begingroup$ @COOLSerdash hmm... so, if I needed a single word for my audience to plug into their "what test did he use" slot without having to explain what a t-statistic is, it sounds like I can use the phrase "Wald-style Test" and move on to the actual point I was trying to make. $\endgroup$ – f1r3br4nd Jun 25 '13 at 17:58
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    $\begingroup$ I'd just use the term "Wald-test". But if that's too complicated, I'd stick with "$t$-test". $\endgroup$ – COOLSerdash Jun 25 '13 at 17:59
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These are t-tests; t-tests aren't only about comparing the means of two groups, it is

any statistical hypothesis test in which the test statistic follows a Student's t distribution if the null hypothesis is supported.

Since $\hat\beta_j$ will be distributed as $t$ when the standard OLS regression assumptions are met (e.g., the errors are normally distributed), the test of $$ \frac{\hat\beta_j-\beta_{j\text{NULL}}}{\text{SE}(\beta_j)} $$ is a t-test.

There is no problem with being only concerned about whether $\hat\beta_j$ is greater than (less than) $\beta_{j\text{NULL}}$; you just use a one-sided t-test.


(Update: @GraemeWalsh makes a reasonable point regarding how to discuss this with people who are not very familiar with statistics.)

I would not name a test in discussing this with people who would not be familiar / would be thrown off by the use of the t-test to assess regression parameters. Instead, I would just say something generic, such as: 'we tested the relationship between $X_j$ and $Y$... '. If someone specifically asked me what kind of test can do that, I would tell them that 'there is a version of the t-test that can be used for this purpose'. I would only explain that "any statistical hypothesis test in which the test statistic follows a Student's t distribution... " is a t-test if they asked further and really wanted to know (in which case, I would probably spend 5 minutes talking with them about these ideas to get them on-board).

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got too long for a comment ... I guess it's an answer

There's lots of information in the output of any typical stats package for regression. Not all the output is necessarily part of your inference. If you're looking at a particular predictor's coefficient, or a particular contrast or whatever, you may be looking at something very different from looking at the regression overall.

If you're saying 'does this model do better than just taking the mean?' then in the usual normal theory test, you're doing an F-test that you could call an analysis of variance, and it's often presented with an analysis of variance table (though such a table isn't produced by summary.lm in R, even though the result of that test is), though I'd normally just say 'an F test of the overall regression'.

If you're looking at a particular coefficient you wouldn't call it an ANOVA; it's a t-test.

There are many ANOVAs, there are many F-tests, there are many t-tests. If I do a nonparametric or robust analysis, they won't even be F- or t- tests, so sometimes I just say something more generic.

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    $\begingroup$ +1, you can get the 'official' ANOVA table in R via anova(). $\endgroup$ – gung Jun 25 '13 at 13:52
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    $\begingroup$ +1 @gung thanks, yes (I pondered whether it was worth mentioning calling anova on lm) though it's perhaps a little more concise than what people might be used to in other packages. A simple example to see one in R for anyone interested is anova(lm(dist~speed,cars)) $\endgroup$ – Glen_b Jun 25 '13 at 23:16
  • $\begingroup$ @gung actually, I'd stay way clear of the default anova (except for partial F-tests comparing two models), and would instead use Anova from the car package. $\endgroup$ – f1r3br4nd Jun 26 '13 at 1:33
  • $\begingroup$ @f1r3br4nd Having just compared anova(lm(dist~speed,cars)) with Anova(lm(dist~speed,cars)) to no fanfare whatever, I presume that some other example is needed to see a large enough disadvantage to justify the strong words 'stay clear'. Why would you stay clear of one and use the other? $\endgroup$ – Glen_b Jun 26 '13 at 1:59
  • $\begingroup$ cat('Simulate y~a+b+a:b, only a:b signif\n'); simlm<-array(dim=c(80,4,5000)); simlm[,3:4,]<-sample(0:1,length(simlm[,3:4,]),rep=T); simlm[,2,]<-rnorm(length(simlm[,2,])); simlm[,1,]<-simlm[,2,]+simlm[,3,]*simlm[,4,]; library(car); cat('Running sims, should take 2-3 min\n'); sim_result<-t(apply(simlm,3,function(ii){ lmii<-lm(ii[,1]~ii[,3]*ii[,4]); unlist(c(summary(lmii)$coef[3,],a1=anova(lmii)[2,],a2=Anova( lmii)[2,],a3=Anova( lmii,type=3)[3,]));})) $\endgroup$ – f1r3br4nd Jun 26 '13 at 2:36

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