# Interaction Models

It is often said that education is the key to a better life. Education makes people more intelligent and better-informed. Education opens up people's minds to new ideas, new experiences, and new opportunities. Education even leads to better health. Education has all these effects because education changes people's very identities. Because you pursued higher education, you are a different person than you would have been. You may not like to admit that attendance at a college or university is changing you, but for better or worse it is. It is impossible to spend several years of your life studying and socializing in an academic setting without being affected in any way. For most people, education leads to a fuller, happier, longer life. Education is also closely associated with income. The education people complete early in life is highly correlated with the wages they earn years later. This is true all over the world, in rich countries and poor countries alike. There are many reasons why more educated people earn higher wages than less educated people. First of all, people who have rich and highly educated parents and tend to receive more education themselves, so more-educated people often start out with more advantages then less-educated people. More importantly, educational credentials are required for admission to many careers, ranging from massage therapists to cancer doctors and everything in between. Education also gives people knowledge and social contacts that help them in their careers. Whatever the reasons, the fact is clear: more highly educated people earn higher wages than less highly educated people. We saw this in Figure 10-8. Though Figure 10-8 focused mainly on the gender gap in wages, it also showed that every extra year of education results in over $3000 more in expected wages for twentysomething Americans. This figure was shown to be robust across all five models estimated. The coefficients for education in Figure 10-8 ranged from a low of 3065 in Model 1 to a high of 3262 in Model 2, with the coefficients for the other models falling somewhere in between these two figures. A coefficient over 3000 means that a four-year college degree is worth more than $12,000 a year in extra wages ($3000 per year x 4 years) to a typical twentysomething American. Unfortunately, as we learned in Figure 10-10 and Figure 10-11, education in America depends strongly on race. This is true in other countries as well, but more detailed data on the subject are available for the United States than for any other country. Among Americans aged 30 and over, Asians are the most educated, followed by Whites, then Blacks, then Others. Model 1 of Figure 10-11 showed that Asians complete on average 0.632 more years of schooling than Whites, while Blacks complete 0.556 fewer years of schooling than Whites. Racial differences overall explain less than 1% of the differences in education between individuals (R2 = .008), but the effects of race are nonetheless statistically significant and substantively important. Figure 11-1 explores the role played by education in explaining racial differences in wage income for Americans aged 30 and over. Data come from the 2008 Survey of Income and Program Participation (SIPP), Wave 2. Model 1 of Figure 11-1 is a straightforward ANOVA model in which wages are regressed on race. The categorical variable "race" has been operationalized using three ANOVA variables: Asian, Black, and Other. The reference group is White. The coefficients for Asian, Black, and Other represent the mean wage differences between people of these races and Whites. The difference is especially large for Blacks. On average, Black Americans aged 30 and over earn $8,746 less than White Americans. Asian Americans earn (on average) $5,991 more than Whites. Combining these two figures, Asian Americans earn (on average) 5991 + 8746 = $14,737 more than Black Americans.
These differences between races are highly significant, but together they explain only a small proportion (juat half a percent) of the total variability in Americans' wages. This doesn't mean that race is unimportant. Around 80% of the American population identifies itself as White (including White Hispanics, such as most Mexican Americans). Since most of the population is White, most of the variability in people's wages is variability among White people. In other words, it is only possible for race to matter for the 20% of the population that is not White (considering White to be the reference group). Even for those people, other factors (like age, education, occupation, industry, place of residence, and employment) may matter more. So it's not surprising that race on its own explains only 0.5% of the total variability in people's wages. The large and significant coefficients in Model 1 indicate that race is important. Since we know from Figure 10-11 that levels of education differ significantly across races in the United States, we might hypothesize that the racial differences in wages observed in Model 1 of Figure 11-1 are due to racial differences in education. After all, if Blacks receive significantly less schooling than Whites, it would not be surprising to find that (on average) they earn lower wages. To investigate this possibility, Model 2 of Figure 11-1 controls for education. Since race is a categorical variable and education is a numerical variable, Model 2 is a mixed model. In Model 2, controlling for education reduces the racial gaps in wage income, but the remaining gaps are still highly significant. Model 2 indicates that Black Americans aged 30 and over and earn (on average) $5,810 less than white Americans of similar education levels. The affect of an additional year of education in Model 2 ($5,269) is almost as large as the wage gap between Blacks and Whites ($5,810). This suggests that one way to address racial differences in wages might be to encourage higher levels of education for Blacks. Right now, Blacks receive on average 0.556 fewer years of schooling than Whites (Model 1 of Figure 10-11). Might raising Black educational levels also raise wages for Blacks? Based on Model 2 of Figure 11-1, the answer seems like it might be yes. On the other hand, it is possible that an additional year of education will raise wages less for Blacks than for Whites. If this happens, it would be very difficult to use education to address the racial gap in wages. Figure 11-2 illustrates the differences between races in the effects of an additional year of education. For Whites, every additional year of education is associated with an increase of $5,393 in wage income. For Asians, the effect is similar, but slightly smaller ($5,298). For Blacks and Others, the economic impact of each additional year of education is much smaller. Black Americans earn on average an additional $4,394 in wage income for each additional year of education, while others earn an additional $4,422 for each year of education.
The results reported in Figure 11-2 are troubling. To begin with, Blacks and Others already have lower education levels on average than Whites and Asians. Then even when Blacks and Others do pursue higher education, they get less benefit from it (in terms of wages) than Whites or Asians. For example, the results reported in Figure 11-2 suggest that a four-year college degree raises White incomes by 5393 x 4 = $21,572 for adults aged 30 and over, while the same college degree raises Black incomes by just 4394 x 4 = $17,576. This is a big difference. On the other hand, we can't tell for sure that the racial differences reported in Figure 11-2 are statistically significant. All the coefficients are significantly different from zero, but that doesn't mean that they are significantly different from each other. It might be that the differences between the coefficients for education in the White model (5393) and the Black mode (4394) are just due to random error in the SIPP data. It seems pretty clear that they are different, but ideally we'd like to know for certain that the difference is statistically significant.
This chapter explains how regression models can be designed to evaluate the significance of group differences in the sizes of regression coefficients. First, special variables called "interaction variables" have to be constructed (Section 11.1). When these are used in regression models they reveal group differences in the slopes associated with independent variables. Second, the inclusion of an interaction variable in a regression model changes the meanings of both the slopes and the intercept for the model (Section 11.2). These changes are most easily understood using graphs to plot the model's regression lines. Third, models with interaction variables can have other control variables just like all other regression models (Section 11.3). In fact, interaction effects are most often found embedded in larger regression models. An optional section (Section 11.4) demonstrates how interaction variables can be calculated using ANOVA variables and applied to many groups at the same time. Finally, this chapter ends with an applied case study of the differences in the economic value of education between the United States and France. (Section 11.5). This case study illustrates how the effects of independent variables can differ significantly across groups of cases. All of this chapter's key concepts are used in this case study. By the end of this chapter, you should understand how to evaluate and interpret between-group differences in regression models.
11.1. Interaction variables Does money set you free? That is to say, does having more money make people feel a greater sense of freedom in life? Not surprisingly, it does. In every country of the world for which data are available, there is a positive correlation between people's sense of personal freedom and their levels of income, though in some countries the relationship is not statistically significant. In the World Values Survey (WVS), personal freedom is measured using the question "How much freedom do you feel?" for which respondents circle a number ranging from 1 "none at all to 10 "a great deal." Mean levels of personal freedom across countries generally run around 7-8 on the scale from 1 to 10, with the poorest people in each country reporting scores that are 1-3 points less than the richest people. Clearly, income matters for people's feelings of freedom, but does it matter the same for all people? In particular, might income matter more for men than for women (or vice versa?). One country where income matters a great deal is Poland. Poland has one of the strongest correlations between income and feelings of freedom anywhere in the world (r = 0.239). Regression models for the impact of income on freedom for men and women in Poland are reported in the first two columns of Figure 11-3. Income is measured by breaking the population up into deciles on a scale from 1 (lowest income) to 10 (highest income). For every one point increase in income, Polish men's freedom ratings rise 0.384 points, while for the same increase in income Polish women's freedom ratings rise just 0.206 points. In Poland, money matters more to men than it does to women.
The male and female regression lines for Poland are plotted in Figure 11-4. The expected values of freedom for women are represented by the solid line while the expected values of freedom for men are represented by the dashed line. At low income levels (left side of the graph) women feel freer than men, while at high income levels men feel freer than women. The two lines cross somewhere between income level 4 and income level 5. For Polish people of middle incomes, men and women report roughly equal levels of personal freedom.
Since the slope for men is 0.384 while the slope for women is just 0.206 points, the difference between the slopes is 0.384 - 0.206 = 0.178. Every additional point of income results in 0.178 less freedom for women than it does for men. Income matters more for men than for women, but is this difference (0.178) statistically significant? In other words, is it possible that the true difference is zero, and that the observed difference (0.178) represents nothing more than random error? After all, if the two lines for men and women were drawn on Figure 11-4 purely at random, it's very unlikely that they would match up exactly. One would almost certainly have a steeper slope than the other. What we want to know is whether or not the observed difference in slopes might have arisen purely at random. To answer this question, it is necessary to set up a regression model in such a way that one of the coefficients represents the difference in slopes between men and women. Then, if this coefficient is significantly different from zero, we can conclude that the difference in the slopes is statistically significant. In such a model, the effects of income would be allowed to interact with a person's gender in such a way as to produce different slopes for income depending on a person's gender. Such models are called interaction models. Interaction models are regression models that allow the slopes of some variables to differ for different categorical groups. Interaction models include (at a minimum) three variables: An independent variable of interest (the variable that is thought to have different slopes for different groups) An ANOVA variable (this can be any 0/1 variable) An interaction variable (equal to the independent variable of interest times the ANOVA variable) The interaction variable is at the heart of the interaction model. Interaction variables are variables created by multiplying an ANOVA variable by an independent variable of interest. In the model for freedom in Poland, the interaction term is computed by multiplying gender (0 for female, 1 for male) by income (scale from 1 to 10). Since gender is zero for all females, the interaction variable is zero for all females (anything times zero is zero). Since gender is one for all males, the interaction variable is the same as income for all males (anything times one is just itself). In concrete terms, the interaction model for freedom in Poland reported in the final column of Figure 11-3 is: Freedom = 5.770 + 0.206 x Income - 0.805 x Gender + 0.178 x Gender x Income For women (gender = 0), this is the same as: Freedom = 5.770 + 0.206 x Income - 0.805 x 0 + 0.178 x 0 x Income Freedom = 5.770 + 0.206 x Income - 0 + 0 Freedom = 5.770 + 0.206 x Income This is the same as the female model reported in Figure 11-3. The interaction model is a little more complicated for men, but not much more. For men (gender = 1), the interaction model is: Freedom = 5.770 + 0.206 x Income - 0.805 x 1 + 0.178 x 1 x Income Freedom = 5.770 + 0.206 x Income - 0.805 + 0.178 x Income Freedom = 5.770 - 0.805 + 0.206 x Income + 0.178 x Income Freedom = 4.965 + 0.384 x Income This is the same as the male model reported in Figure 11-3. If the interaction model just gives the same two models we started with, why run the interaction model at all? One reason is that the interaction model uses all the data (N = 903 cases) all in one model instead of separating the data out into male and female models. Another, much more important reason is that the interaction model tells us the statistical significance of the interaction variable. The coefficient of the interaction variable represents the difference in slopes between the two groups in the model, in this case between men and women. Notice how the coefficient of the interaction variable in the interaction model in Figure 11-3 is 0.178, exactly equal to the difference between the slope for income in the male model and the slope for income in the female model. The interaction model tells us that this coefficient is statistically significant. As a result, we can conclude that the slope of the relationship between income and freedom is significantly higher for men than for women. Income is significantly more important for Polish men than for Polish women in promoting feelings of personal freedom.
11.2. Slopes and intercepts in interaction models In Figure 11-3, it's no coincidence that the results of the male and female models can be calculated from the interaction model. The interaction models is based on the same data and variables as the other models. The only real difference between the interaction models and the two separate models for men and for women is their purpose. The male model is used to evaluate the slope of the relationship between income and freedom for men. The female model is used to evaluate the slope of the relationship between income and freedom for women. The interaction model is used to evaluate the difference between the slope for men and the slope for women. Conveniently, the interaction model can also be used to find out the slope for men and the slope for women, so in the end only one model is necessary. The slope for income reported in the interaction model is the slope for income for the reference group. In the interaction model in Figure 11-3, the reference group is female (gender = 0). The slope for income for women is the main effect reported in the interaction model. Main effects are the coefficients of the independent variable of interest in an interaction model for the reference group. The difference in slopes between women and men in Figure 11-3 is 0.178 (the slope for men is 0.178 points steeper than the slope for women). This is the coefficient of the interaction variable (gender x income) in Figure 11-3. The coefficient of the interaction variable in an interaction model is called an interaction effect. Interaction effects are the coefficients of the interaction variables in an interaction model. When the interaction effect is statistically significant it means that there is a significant difference in slopes between the two groups. The interaction model in Figure 11-3 includes one more variable, gender. The coefficient for gender in the interaction model represents the difference between the intercept of the regression line for men and the regression line for women. This is called an intercept effect. Intercept effects are the coefficients of the ANOVA variables in an interaction model. In the regression of freedom on income for men, the intercept was 4.965, meaning that a man with zero income would be expected to report a personal freedom level of 4.965 on a scale from 1 to 10. In the regression of freedom on income for women, the intercept was 5.770, meaning that a woman with zero income would be expected to report a personal freedom level of 5.770 on a scale from 1 to 10. The intercept effect (-0.805) is the difference between the intercept for men and the intercept for women: 4.965 - 5.770 = -0.805. In interpreting interaction models, we're usually not interested in intercept effects, and they're usually ignored. It's necessary for the ANOVA variable to be included in the model though (like gender in Figure 11-3). Without it, interaction models produce meaningless results. In Poland, we found that income mattered much more to men than to women in determining their feelings of personal freedom. In other countries the situation might be different. Figure 11-5 reports the results of a set of models that are identical to those estimated in Figure 11-3, but this time the models are estimated using data from Australia. For Australian men (male model), every one-point rise in the scale of incomes is associated with a 0.061 point rise in feelings of personal freedom. For Australian women (female model) the associated rise is 0.143 points -- more than twice as much. For both men and women the association between income and freedom is statistically significant, but it is much more significant for women than for men.
The third model reported in Figure 11-5 is the interaction model. This model includes the independent variable of interest (income), a categorical ANOVA variable (gender), and an interaction variable (gender x income). In the interaction model, the main effect of income is 0.143, which is identical to the coefficient for income in the female model. This main effect represents the impact of a one-point increase in income on women's expected feelings of personal freedom. The interaction effect is -0.083. This is the difference between the slope for men and the slope for women in the relationship between income and freedom: 0.061 - 0.143 = -0.083. The interaction effect is statistically significant, indicating that the slope for men is significantly shallower than the slope for women. The intercept effect reported in Figure 11-5 is 0.393, but this is not of any particular theoretical interest. The statistical results reported in Figure 11-5 are graphed in Figure 11-6. The results of interaction models can be difficult to visualize, but plotting them out usually makes them very clear. Figure 11-6 shows that the relationship between income and freedom is weaker for Australian men than it is for Australian women. Both lines rise with income, but the line for women rises faster. Poor women experience less personal freedom than poor men, but rich women experience greater personal freedom than rich men.
Interaction effects are not always significant. In fact, it can be difficult to find significant interactions. For example, in the United States the gender differences in the importance of income are not statistically significant. The results of freedom versus income regressions for the United States are reported in Figure 11-7. The slopes for both men and women are highly significant, but the interaction effect examining the difference in slopes is not.
Figure 11-8 represents the United States results graphically. The two regression lines (women and men) are nearly parallel. The graph seems to show an intercept effect (the line for women is higher than the line for men), but the model results show that this difference is also not significant, though in any case it is not of theoretical interest. In the United States, rich people feel significantly freer than poor people, but there is no evidence of differences between women and men in the importance of income for freedom. For both women and men, higher income makes Americans feel more free in equal measure.
11.3. Interaction effects in mixed models with control variables The results reported in Figure 11-2 showed that every additional year of education raises wages for Whites more than it does for Asians, Blacks, and Others. If education has a different impact on wages for people of different races, might is also have a different impact on wages for people of different genders? In Figure 10-8 it was shown that every extra year of education results in over $3000 more in expected wages for twentysomething Americans. The main results of Figure 10-8 are reprinted as Model 1 in Figure 11-9. Figure 11-9 uses the same 2008 SIPP Wave 2 data to regress wage income on a number of predictors, including gender (coded as 0 for men and 1 for women). Model 1 of Figure 11-9 shows that twentysomething American women earn, on average, $6,591 less than twentysomething American men (after controlling for age, rage, Hispanic status, and education).
Model 2 of Figure 11-9 introduces an interaction variable, Education x Female. In this new model, the main effect of Education is now 3114, indicating that for men (Female = 0) every additional year of education increases the expected value of wages by $3114 The coefficient of the interaction variable (Education x Female) is 312, indicating that for every additional year of education women can expect to receive $312 more than men do. Since men receive $3114 (on average) for every year of education, this means that women receive (on average) 3114 + 312 = $3326 more in wages. The positive interaction effect of 312 means that education actually helps women more than it helps men. This suggests that education could be effective in reducing the gender gap in wages. On the other hand, the interaction effect in Model 2 is not statistically significant. This means that the positive result might have occurred at random. In Model 2 the intercept effect (the coefficient for the variable Female) is -10857. Unlike in Model 1, this coefficient doesn't directly say anything about the differences in wages between women and men. The intercept effect in an interaction model is not very meaningful from the standpoint of interpreting results (though it does have to be included in the model). In order to make conclusions about the overall differences in wages between men and women, you would have to look at the coefficient for Female in Model 1, which doesn't include any interaction effects. Since the interaction between education and gender is not significant, Model 3 in Figure 11-9 examines a different interaction variable: the interaction between gender and age (Age x Female). This interaction is highly significant. As women get older, their incomes lag farther and farther behind men's incomes. The main effect for age in Model 3 (2430) indicates that American men are expected to earn $2430 more every year throughout their twenties. The interaction effect for age (Age x Female) of -786 implies that women's incomes go up by $786 less than men's incomes for every year they get older. In other words, while men's expected wages rise by $2430 a year, women's expected wages rise by 2430 - 786 = $1644 a year. This is a large and statistically significant difference (probability < .001). Aging is incredibly more remunerative for men than it is for women. There's no reason why both interaction variables can't be included in the same model. This is done in Model 4. In Model 4, both interaction effects are significant, but the interaction for age is much more so. In Model 5, additional control variables are included alongside the interaction variables. Even after controlling for marriage, children, full-time employment status, school enrolment status, and industry of employment, the interaction effect for age remains highly significant. On the other hand, the interaction effect for education is reduced to a very small and insignificant figure (less then $100 per year of education). The results reported in Model 5 suggest that additional education will not do much to reduce the gender gap in wages between men and women in America. Instead, the gap tends to widen further and further as men and women age, at least over the course of their twenties. As Figure 11-9 illustrates, interaction effects can be used in mixed models with other numerical and categorical independent variables. Multiple interaction effects can even be estimated within a single model. The interpretations of these models are no different from the interpretation of ordinary regression and interaction models. Any model that includes both numerical and ANOVA variables can include one or more interactions as well.
11.4. Multiple categorical interactions (optional/advanced) If a regression model can include multiple interactions for the same ANOVA variable (education and age by gender in Figure 11-9), it stands to reason that a regression model can include the same interaction for multiple ANOVA variables. In Figure 11-2 it was shown that the expected effects of education on wages were different for Americans of different races. The impacts ranged from a low of $4394 in extra wages per year of education for Black Americans aged 30 and over to a high of $5993 in extra wages per year of education for White Americans. Figure 11-10 shows how an interaction model can be used to examine the statistical significance of these interracial differences. Model 1 of Figure 11-10 includes three ANOVA variables for race (White is the reference group), the independent variable of interest (Education), and three interaction variables for Race x Education.
The main effect of Education in Model 1 is 5393, confirming that every additional year of education is associated with an increase of $5393 in annual wages for White Americans aged 30 and over. The three interaction effects are all negative, confirming that for all other races education increases wages less than it does for whites. For Asians, the interaction effect is small and not statistically significant: education raises wages for Asians at roughly the same rate as it does for Whites. While impact of each additional year of education is $5393 for Whites, it is 5393 - 96 = $5297 for Asians (the difference between this result and that reported in Figure 11-2 is due to rounding). The interaction effects for Blacks and Others are negative and statistically significant (in the case of Blacks, highly significant). This means that the effects of each extra year of education on wages are significantly lower for Blacks and Others than it is for Whites. One shortcoming of Model 1 is that (as with any ANOVA model) all of the group differences are examined relative to the reference group (in this case, Whites). So, for example, Model 1 tells us whether or not the slope for Others differs significantly from the slope for Whites (it does), but it doesn't tell us whether or not the slope for the slope for Others differs significantly from the slope for Blacks. In order to find the statistical significance of the differences in slope between Blacks and the other three races, a new model has to be estimated that uses Blacks as the reference group. This is done in Model 2, in which the same variables are organized in such a way as to highlight the significance of differences from Blacks considered as a reference group (rather than Whites). Since Model 2 contains the same information as Model 1, the R2 doesn't change from Model 1, but the coefficients (and their significance levels) do. In Model 2 of Figure 11-10, the main effect of Education is now 4394, consistent with the fact that Black is now the reference group and every additional year of education is associated with an extra $4394 in expected annual wages for Black Americans aged 30 and over. The coefficient for the White interaction is 999, indicating that Whites are expected to earn an extra 4394 + 999 = $5395 for each additional year of schooling, just as in Model 1. What Model 2 tells us that Model 1 does not is that the difference between the impact of education for Others versus Blacks ($28 per year of education) is not statistically significant. It also tells us that the difference in the returns to education for Asians versus Blacks ($904 per year of education) is statistically significant. Model 1 and Model 2 represent the same information, but by using different reference groups they allow the examination of the significance of different racial differences. The expected values of wages based on education for all four races are plotted in Figure 11-11. The values plotted in this graph could be calculated from either of the two regression models (or from the models reported in Figure 11-2) Notice how the slope for Whites is steeper than the slopes for the other three races. Among high school dropouts, Whites are expected to earn barely more than Others, but among people with post-graduate education, Whites are expected to earn almost as much as Asians. The differences in slopes between Asians and Whites on the one hand and Blacks and Others on the other mean that racial inequalities in American wages increase as education levels rise. Rising education levels in society are not likely to make American society more equal. Quite the contrary: they seem likely to make existing racial inequalities even worse.
11.5. Case study: Cross-national comparison of democracy ratings In Chapter 5 it was argued that older people in Taiwan might rate the quality of Taiwan's democracy more highly because older Taiwanese had lived through a period of dictatorship before 1991. This might be called the "dictatorship theory" of democracy ratings. It was suggested in Chapter 5 that younger people who came of age in the democratic era might be more demanding, and as a result be less satisfied with Taiwan's democracy than older people. A regression model confirmed this reasoning. Taiwanese people's ratings of the democratic quality of their government was found to rise with age. The fact that Taiwanese people's ratings of Taiwanese democracy rise with age tends to support the dictatorship theory, but it doesn't prove the theory. An alternative theory, the "youth theory," might be that young people everywhere are more demanding when it comes to democracy, while older people are less demanding. One way to shed light on the relative merits of the two theories would be to compare the relationship between age and democracy ratings in Taiwan to those in another country. For this purpose, the United States has been chosen as a reference country, since the United States has never experienced a period of dictatorship. As a reminder, people's ratings of democracy have been scored on a scale from 0 to 100 where: Rating = 0 means the respondent thinks there is not enough democracy in her or his country Rating = 50 means the respondent thinks there is just the right amount of democracy in her or his country Rating = 100 means the respondent thinks there is too much democracy in her or his country The mean democracy rating (for all ages) was 38.7 in Taiwan and 37.2 in the United States. Regression models reporting the relationship between age and democracy ratings for the United States and Taiwan are reported in Figure 11-12. The Taiwan model is identical to that reported in Figure 5-7, except that now we have significance levels and an R2 to work with in addition to just the coefficients. The increase in democracy ratings with age is, in fact, highly statistically significant. The United States model, on the other hand, shows that democracy ratings decline with age in the United States. The decline with age in the US is even stronger than the rise with age in Taiwan. This seems to support the dictatorship theory over the youth theory of democracy ratings.
A formal interaction model of the differences in the slope of the relationship between age and democracy ratings is reported in the final column of Figure 11-12. This model includes the independent variable of interest (Age), the ANOVA variable that distinguishes between the two groups (Country), and an interaction variable (Country x Age). The main effect of Age in the interaction model is -0.117, which is the effect of age on democracy ratings in the United States. The interaction effect for age is 0.222, indicating that the coefficient for Age in Taiwan is 0.222 points higher than the coefficient for Age in the United States. This difference in the slope of the relationship between age and democracy ratings is highly significant (probability < .001). The intercept effect (-8.680) is of no theoretical interest in this model, though it is used when computing expected values. Those expected values of democracy ratings for people of different age in Taiwan versus the United States are plotted in Figure 11-13. Obviously, democracy ratings rise with age in Taiwan, while they decline with age in the United States. This tends to confirm the dictatorship theory that older Taiwanese might think less highly of Taiwan's democracy had they not themselves experienced what life was like under a dictatorship. Whether the differences between Taiwan and the United States are due to this reason or some other reason, it is very clear that the relationship between age and democracy ratings is different in the two countries. There is a significant interaction between country and the effects of age on people's democracy ratings.

## Chapter 11 Key Terms

* Interaction effects are the coefficients of the interaction variables in an interaction model.
* Interaction models are regression models that allow the slopes of some variables to differ for different categorical groups.
* Interaction variables are variables created by multiplying an ANOVA variable by an independent variable of interest.
* Intercept effects are the coefficients of the ANOVA variables in an interaction model.
* Main effects are the coefficients of the independent variable of interest in an interaction model for the reference group.
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