# Analytical Treatment of One-Dimensional Markov Processes. by Petr Mandl

By Petr Mandl

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3) The statistical assumptions of the linear regression model concern the distribution of ε. d N (0, σε2 ), meaning that ε is independently and identically distributed and has a • • • • normal distribution mean zero at every value of age constant variance σε2 at every value of age values that are statistically independent. In Sect. 7 we will see that the ﬁrst assumption may sometimes be relaxed. 3), and Fig. 1; violations can be examined and repaired using methods also introduced in Sect. 7. The third assumption, of constant variance, is sometimes called homoscedasticity; data which violate this assumption are called heteroscedastic, and can be dealt with using methods also discussed in Sect.

Suppose, for example, we wished to treat behavior pattern as the outcome variable and weight as the predictor. We might ﬁrst divide weight into four categories: ≤140 pounds, >140–170, >170–200, and >200. As with histograms, we need enough categories to avoid 2 Exploratory and Descriptive Methods 100 Systolic Blood Pressure 150 200 250 22 A1 A2 B3 B4 Behavior Pattern Fig. 10. Boxplots of Systolic Blood Pressure by Behavior Pattern loss of information, without deﬁning categories that include too few observations.

However, linear regression models are commonly used with outcomes that are at best approximately normal, even after transformation. Fortunately, in large samples the t-tests and conﬁdence intervals for βˆ0 and βˆ1 are valid even when the underlying outcome is not normal. How large a sample is required depends on how far and in what way the outcome departs from normality. If the outcome is uniformly distributed, meaning that every value in its range is equally likely, then the t-tests and conﬁdence intervals may be valid with as few as 30–50 observations.