A previous article discusses the definitions of three kinds of moments for a continuous probability distribution: raw moments, central moments, and standardized moments. These are defined in terms of integrals over the support of the distribution. Moments are connected to the familiar shape features of a distribution: the mean, variance,
The noncentral t distribution is a probability distribution that is used in power analysis and hypothesis testing. The distribution generalizes the Student t distribution by adding a noncentrality parameter, δ. When δ=0, the noncentral t distribution is the usual (central) t distribution, which is a symmetric distribution. When δ >
An integer can be represented in many ways. This article shows how to represent a positive integer in any base b. The most common base is b=10, but other popular bases are b=2 (binary numbers), b=8 (octal), and b=16 (hexadecimal). Each base represents integers in different ways. Think of a
The SELECT-WHEN statement in the SAS DATA step is an alternative to using a long sequence of IF-THEN/ELSE statements. Although logically equivalent to IF-THEN/ELSE statements, the SELECT-WHEN statement can be easier to read. This article discusses the two distinct ways to specify the SELECT-WHEN statement. You can use the first
It isn't easy to draw the graph of a function when you don't know what the graph looks like. To draw the graph by using a computer, you need to know the domain of the function for the graph: the minimum value (xMin) and the maximum value (xMax) for plotting
A colleague was struggling to compute a right-tail probability for a distribution. Recall that the cumulative distribution function (CDF) is defined as a left-tail probability. For a continuous random variable, X, with density function f, the CDF at the value x is F(x) = Pr(X ≤ x) = ∫
I previously wrote about partial leverage plots for regression diagnostics and why they are useful. You can generate a partial leverage plot in SAS by using the PLOTS=PARTIALPLOT option in PROC REG. One useful property of partial leverage plots is the ability to graphically represent the null hypothesis that a
A previous article shows how to compute the probability density function (PDF) for the multivariate normal distribution. In a similar way, you can compute the density function for the multivariate t distribution. This article discusses the density function for the multivariate t distribution, shows how to compute it, and visualizes
Recently, I needed to solve an optimization problem in which the objective function included a term that involved the quantile function (inverse CDF) of the t distribution, which is shown to the right for DF=5 degrees of freedom. I casually remarked to my colleague that the optimizer would have to
A common question on SAS discussion forums is how to randomly assign observations to groups. An application of this problem is assigning patients to cohorts in a clinical trial. For example, you might have 137 patients that you want to randomly assign to three groups: a control group, a group
