Thursday, October 7: Interactions and Ordered Categorical Variables I. Interpreting Interaction Effects (also see Greene, pp. 123-124) A. The Concept of an Interaction. Remember that interactions measure how much changes on one variable affect the effect of another variable. Put another way, they reveal whether or not the effect of variable changes in different contexts. For instance, increases in ethnolinguistic fractionalization could lead to significant declines in political stability under authoritarian regimes, but lead to increases in the stability of democracies. If this is true, there will be a strong interaction between type of political system and ethnolinguistic fractionalization. B. Measuring an Interaction Effect. To test the hypothesis that Variable x1 influences the effect of Variable x2 (and vice-versa), your model should include an interaction term that is the product of these two variables. If the effect of one variable is altered by the other, the coefficient on this interaction term (x1x2) will be significant. In ordinary least squares regression, evaluating the effect of x1 (ELF fractionalization in this example) at any particular level of x2 (type of political system) is straightforward: you add the coefficient of x1 to the product of the coefficient on x1x2 and the particular value of x2 in which you are interested. For the model: y = β0 + β1x1+ β2x2+ β3x1x2 + e We can get a predicted value by yhat = β0 + β1x1+ β2x2+ β3x1x2 And we can obtain the first difference for x1, the change brought in yhat by a one-unit change in x1 from x1low to x1high while all other factors are held constant, by: Δy = β0 + β1x1high + β2x2+ β3x1highx2 - β0 - β1x1low - β2x2 - β3x1lowx2 Δy = β1x1high - β1x1low + β3x1highx2 - β3x1lowx2 Δy = β1(x1high - x1low) + β3x2(x1high - x1low) Δy = β1 + β3x2 C. Interaction Effects in Maximum Likelihood Models. If only life were so simple in the ML world. In OLS, this expression reduces so nicely because the predicted value yhat for a given observation is just a linear function of the coefficients and explanatory variables. The contribution of β2x2 to yhat is not going to vary across observations, so it can drop out of the expression. But in ML models with more complex systematic components, β2x2 and all of the other terms will be put through a transformation such as the logit transformation. In a model with an interaction term, the effect of x2 will depend upon both x1 and upon the levels of all the other variables in the equation (which determine where this observation is on the logit curve when the shift in x1 begins). Just as we did when we were finding simple first differences, we will have to pick some value to set all other variables constant at. One pitfall to avoid is that when you tell Clarify to setx mean, it will set the interaction term constant at the mean of the interaction term (the product of x1 and x2). This is not very substantively meaningful, since the interaction term isn’t a real variable. Instead of setting it constant at the mean of the product of the two variables, we should set it constant at the product of the means of x1 andx2. Then, to find the effect of a one-unit shift in x1 at some particular value of x2, we simulate the first difference brought by both a one-unit change in x1 (analogous to β1) and the change in the interaction term that results from a one-unit change in x1 (analogous to β3x2). D. Syntax for Interactions in Clarify. Suppose I want to predict whether Senate committees in the 50 states are required to report legislation that they are assigned. I begin with a model that attempts to explain the presence of a reporting rule (senhear=1) by using the three components of legislative professionalism, the house’s session length, staffing levels, and salaries, along with a measure of the “moralism” of the state’s political culture. I run the following logit model, using Clarify. . estsimp logit senrepor totalday salary staffup moral Iteration 0: log likelihood = -26.345398 Iteration 1: log likelihood = -20.849193 Iteration 2: log likelihood = -20.011675 Iteration 3: log likelihood = -19.917045 Iteration 4: log likelihood = -19.914806 Iteration 5: log likelihood = -19.914805 Logit estimates Number of obs = 50 LR chi2(4) = 12.86 Prob > chi2 = 0.0120 Log likelihood = -19.914805 Pseudo R2 = 0.2441 ------------------------------------------------------------------------------ senrepor | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- totalday | -.0141113 .0099893 -1.41 0.158 -.0336899 .0054674 salary | .0000921 .0000477 1.93 0.053 -1.33e-06 .0001856 staffup | -2.492502 1.392717 -1.79 0.074 -5.222177 .2371723 moral | 2.617587 1.120754 2.34 0.020 .4209504 4.814224 _cons | -2.046383 .9856179 -2.08 0.038 -3.978159 -.1146075 ------------------------------------------------------------------------------ Simulating main parameters. Please wait.... % of simulations completed: 20% 40% 60% 80% 100% Number of simulations : 1000 Names of new variables : b1 b2 b3 b4 b5 Now I can look at the effects increasing the salary level from Maine’s value ($12,900) to California’s ($99,250), holding other variables constant at their means: setx mean simqi fd(pr) changex(salary 12900 99250) First Difference: salary 12900 99250 Quantity of Interest | Mean Std. Err. [95% Conf. Interval] ---------------------------+-------------------------------------------------- dPr(senrepor = 0) | -.842876 .2485586 -.9939116 -.0548535 dPr(senrepor = 1) | .842876 .2485586 .0548535 .9939116This tells me that moving from Maine’s salary to California’s salary, all other factors being equal, increase the chances of requiring Senate committees to report bills by 0.84, with a confidence interval of (0.05 to 0.99). Now suppose I want to test whether the effect of salary is different in moralistic and nonmoralistic states, and estimates those two