I was recently asked about when to use one and two tailed tests.
The long answer is: Use one tailed tests when you have a specific hypothesis about the direction of your relationship. Some examples include you hypothesize that one group mean is larger than the other; you hypothesize that the correlation is positive; you hypothesize that the proportion is below .5.
The short answer is: Never use one tailed tests.
Why?
1. Only a few statistical tests even can have one tail: z tests and t tests. So you’re severely limited. F tests, Chi-square tests, etc. can’t accommodate one-tailed tests because their distributions are not symmetric. Most statistical methods, such as regression and ANOVA, are based on these tests, so you will rarely have the chance to implement them.
2. Probably because they are rare, reviewers balk at one-tailed tests. They tend to assume that you are trying to artificially boost the power of your test. Theoretically, however, there is nothing wrong with them when the hypothesis and the statistical test are right for them.
A great article for specifying Mixed Models in SAS:
Mixed up Mixed Models
by Robert Harner & P.M. Simpson
Anyone doing mixed modeling in SAS should read this paper, originally presented at SUGI: SAS Users Group International conference. It compares the output from Proc Mixed and Proc GLM when specified different ways. There are some subtle distinctions in the meaning of the defaults in the Repeated and Random statements, and this paper does an excellent job of clarifying them.
I just wanted to follow up on my last post about Regression without Intercepts.
Regression through the Origin means that you purposely drop the intercept from the model. When X=0, Y must = 0.
The thing to be careful about in choosing any regression model is that it fit the data well. Pretty much the only time that a regression through the origin will fit better than a model with an intercept is if the point X=0, Y=0 is required by the data.
Yes, leaving out the intercept will increase your df by 1, since you’re not estimating one parameter. But unless your sample size is really, really small, it won’t matter. So it really has no advantages.
One of the main assumptions of linear models such as linear regression and analysis of variance is that the residual errors follow a normal distribution. To meet this assumption when a continuous response variable is skewed, a transformation of the response variable can produce errors that are approximately normal. Often, however, the response variable of interest is categorical or discrete, not continuous. In this case, a simple transformation cannot produce normally distributed errors.
A common example is when the response variable is the counted number of occurrences of an event. The distribution of counts is discrete, not continuous, and is limited to non-negative values. There are two problems with applying an ordinary linear regression model to these data. First, many distributions of count data are positively skewed with many observations in the data set having a value of 0. The high number of 0’s in the data set prevents the transformation of a skewed distribution into a normal one. Second, it is quite likely that the regression model will produce negative predicted values, which are theoretically impossible.
An example of a regression model with a count response variable is the prediction of the number of times a person perpetrated domestic violence against his or her partner in the last year based on whether he or she had witnessed domestic violence as a child and who the perpetrator of that violence was. Because many individuals in the sample had not perpetrated violence at all, many observations had a value of 0, and any attempts to transform the data to a normal distribution failed.
An alternative is to use a Poisson regression model or one of its variants. These models have a number of advantages over an ordinary linear regression model, including a skew, discrete distribution, and the restriction of predicted values to non-negative numbers. A Poisson model is similar to an ordinary linear regression, with two exceptions. First, it assumes that the errors follow a Poisson, not a normal, distribution. Second, rather than modeling Y as a linear function of the regression coefficients, it models the natural log of the response variable, ln(Y), as a linear function of the coefficients.
The Poisson model assumes that the mean and variance of the errors are equal. But usually in practice the variance of the errors is larger than the mean (although it can also be smaller). When the variance is larger than the mean, there are two extensions of the Poisson model that work well. In the over-dispersed Poisson model, an extra parameter is included which estimates how much larger the variance is than the mean. This parameter estimate is then used to correct for the effects of the larger variance on the p-values. An alternative is a negative binomial model. The negative binomial distribution is a form of the Poisson distribution in which the distribution’s parameter is itself considered a random variable. The variation of this parameter can account for a variance of the data that is higher than the mean.
A negative binomial model proved to fit well for the domestic violence data described above. Because the majority of individuals in the data set perpetrated 0 times, but a few individuals perpetrated many times, the variance was over 6 times larger than the mean. Therefore, the negative binomial model was clearly more appropriate than the Poisson.
All three variations of the Poisson regression model are available in many general statistical packages, including SAS, Stata, and S-Plus.
References:
- Gardner, W., Mulvey, E.P., and Shaw, E.C (1995). “Regression Analyses of Counts and Rates: Poisson, Overdispersed Poisson, and Negative Binomial Models”, Psychological Bulletin, 118, 392-404.
- Long, J.S. (1997). Regression Models for Categorical and Limited Dependent Variables, Chapter 8. Thousand Oaks, CA: Sage Publications.
With SAS, it’s almost always the semicolons!
I read that recently–I can’t remember where now (if you wrote it, let me know–I’ll link!).
I spent the day at Cornell doing SAS programming; I kept expecting Andy Bernard to show up.
Anyway, I was reminded of that quote because, as you guessed it, it was almost always that I forgot a semicolon when I got a SAS error. The errors never mention a semicolon, but that is always the first thing to check for.
I was also reminded today that another common mistake is to forget the SET statement when creating a new data set. If SAS tells you your variables don’t exist, but you know they do, it’s because you forgot the SET statement.
I was also reminded of my best SAS time saver ever. F12: output;cle;log;cle;wpgm. At first I set it up wrong and was lost for a few minutes without it. Luckily I discovered my mistake pretty quickly.
