Distribution Analysis: Distributional Modeling

The Plots Tab

You can use the Plots tab to create the following plots:

Histogram with density estimators

creates a histogram overlaid with density curves for the parametric distributions specified on the Estimators tab.

Quantile-quantile plots

creates one Q-Q plot for each parametric distribution specified on the Estimators tab.

Empirical cumulative distribution function (CDF)

creates a plot of the empirical cumulative distribution function.

Note: Stat Studio adds a density curve to an existing histogram when both of the following conditions are satisfied:

• The histogram is the active window when you select the analysis.
• The histogram variable and the analysis variable are the same.

Q-Q Plots

A Q-Q plot graphically indicates whether there is agreement between quantiles of the data and quantiles of a theoretical distribution. If the quantiles of the theoretical and data distributions agree, the plotted points fall along a straight line. For most distributions, the slope of the line is the value of the scale parameter, and the intercept of the line is the value of the threshold or location parameter. (For the lognormal distribution, the slope is , where is the value of the scale parameter.) The parameter estimates for the distribution that best fits the data appear in an inset in the Q-Q plot.

Table 15.1 presents reasons why the points in a Q-Q plot might not be linear.

Table 15.1: Interpretation of Q-Q Plots

Description of Point Pattern	Possible Interpretation
All but a few points fall on a line	Outliers in the data
Left end of pattern is below the line; right end of pattern is above the line	Long tails at both ends of the data distribution
Left end of pattern is above the line; right end of pattern is below the line	Short tails at both ends of the data distribution
Curved pattern with slope increasing from left to right	Data distribution is skewed to the right
Curved pattern with slope decreasing from left to right	Data distribution is skewed to the left
Most points are not near line with scale parameter and location parameter	Data do not fit the theoretical distribution

Caution: When the variable being graphed has repeated values, the Q-Q plot produced by Stat Studio is different from the Q-Q plot produced by the UNIVARIATE procedure. The UNIVARIATE procedure arbitrarily ranks the repeated values and assigns a quantile for the theoretical distribution based on the ranks. Two observations with the same value are assigned different quantiles. If a variable has many repeated values, the Q-Q plot produced by the UNIVARIATE procedure looks like a staircase. However, Stat Studio (and SAS/INSIGHT) averages the ranks of repeated values. Two observations with the same value are assigned the same quantiles for the theoretical distribution.

CDF Plots

A CDF plot shows the empirical cumulative distribution function. You can use the CDF plot to examine relationships between data values and data proportions. For example, you can determine whether a given percentage of your data is below some upper control limit. You can also determine what percentage of the data has values within a given range of values.

The inset for the CDF plot displays two statistics. The first is the number of nonmissing observations for the plotted variable. The second is labeled . If is the 95% quantile for Kolmogorov's distribution () and is the number of nonmissing observations, then (D'Agostino and Stephens 1986)

The 95% confidence limits in the CDF plot are obtained by adding and subtracting from the empirical CDF. They form a confidence band around the estimate for the cumulative distribution function.