As with our other tests, you can assess the significance of a coefficient by looking at its p-value. While our critical value of .05 is somewhat arbitrary, output in R will mark tests that are statistically significant at the .05 level with at least one *. 

Remember: tests with p values of greater than .05 do not provide enough evidence to conclude that there is or is not a relationship or effect. 

Note: Be careful when interpreting the magnitude and direction for insignificant coefficients. 

Note: We often designate significant effects using parentheticals at the end of the sentence (i.e., p < .05)

The intercept can be understood as the expected or predicted value of Y given that X is 0. Note that the substantive interpretation will depend upon if the IV is quantitative or qualitative.

Consider the example from a dig site showing the relationship between dinosaur's femur length and body mass. 

For a one cm increase in femur length, we expect body mass to increase by 49 KG, holding all else constant (p < .05)

The expected weight for a dinosaur whose femur is 0 cm long is 255 kgs. 

Specifying the reference is especially important if the independent, qualitative variable has more than 2 categories.

Relative to those whose major is in the business school, the employer-rated people skills among those whose major is in the engineering school is rated 2.25 points lower.  

Relative to those whose major is in the business school, the employer-rated people skills among those whose major is in the humanities school is rated 1.56 points higher.  

Employers rate business school majors' people skills as 5.41.

When quantitative independent variables are standardized, they are transformed to have a mean of 0 and a standard deviation of 1, which makes them more comparable to other standardized coefficients.

Standardized regression coefficients represent the change in the dependent variable that occurs when the independent variable increases by one standard deviation, while holding all other independent variables constant. 

If Engineering majors are rated 2.25 points lower than Business Majors, and Humanities Majors are rated 1.56 points higher, then those whose major was in the business school are rated 3.81 points lower than those who majored in the humanities (2.25+1.56). 

However, note that we don't have a significance test for this relationship. This is one reason among many to think carefully about the reference category! 

Whether or not the intercept needs to or can be substantively interpreted depends upon the model at hand. There are many perspectives on intercepts, but suffice it to say that you should take care when interpreting intercepts - especially for non-specialists.

Think carefully when substantively interpreting intercepts.  

Does the confidence interval include the possibility that the population parameter for the slope is 0?

We use statistics to generalize from the sample regression line (our predicted equation line) to a theoretical population regression line. 

The confidence interval of the estimate represents a range of potential regression lines that we are to some degree confident includes the population regressioon line. 

R uses this t-value to determine the p value. 

When the relationship between X and Y is not linear, you can transform the Y or X variable to better match the relationship.  

How do we determine which transformation will best fit a non-linear relationship? Look at the residuals. 

Mathematically, interpreting models with non-linear relationships is the same. However, it is substantively more complicated because knowing whether an increase in X will lead to an increase or decrease (and to what extent) becomes dependent on much more context.  

For this class, it is best to examine a plot, create hypothetical examples and calculate the expected values using observed values that make sense, and use deductive reasoning.

Note: Some non-linear transformations change the interpretation of the intercept, but that change is beyond the scope of this class. 

In the social sciences, it is common for a relationship to be best represented by a polynomial function. Simply, polynomials include more than one term that each contains or represents the same variable. 

expected income = 6867+1135(age)-11(age*age)

Polynomials can help to model many curvilinear relationships - not just the inverse U shape shown on the left. 

In the log-linear model, we predict the natural log of the dependent variable. 

When the dependent variable is logged, we are predicting a percent change in Y.

If the independent variable is logged, then we are predicting the expected change in Y in its native unit given a 1 percent increase in X.

If both the independent and dependent variables are logged, then we are predicting a percent change in Y given a percent change in X. 

Logged models describe a model where all variables are logged.