β1: partial regression slope, change in Y per unit change in X1 holding other X-vars constant
β2: partial regression slope, change in Y per unit change in X2 holding other X-vars constant
βp: partial regression slope, change in Y per unit change in Xp holding other X-vars constant
What is partial - the relationship between a predictor variable and the response variable while holding all other predictor variables constant. It tells you the isolated effect of one variable, controlling for the influence of others.
Lecture 10: Partial regression slopes
model: ant_spp ~ elevation + latitude
The partial slope for elevation tells you how ant species richness changes with elevation when latitude is held constant
1-m increase in elevation - ant spp decrease by 0.012 spp, holding latitude constant
Or 100-meter increase in elevation, lose 1.2 spp
The partial slope for latitude tells you how ant species richness changes with latitude when elevation is held constant
1-degree increase in latitude, ant spp decrease by 2 species, holding elevation constant
Moving north = fewer ant spp, even when comparing sites at same elevation
model <-lm(ant_spp ~ elevation + latitude, data = ant_df)summary(model)
Call:
lm(formula = ant_spp ~ elevation + latitude, data = ant_df)
Residuals:
Min 1Q Median 3Q Max
-6.1180 -2.3759 0.3218 1.9070 5.8369
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 98.49651 26.50701 3.716 0.00147 **
elevation -0.01226 0.00411 -2.983 0.00765 **
latitude -2.00981 0.61956 -3.244 0.00427 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.022 on 19 degrees of freedom
Multiple R-squared: 0.5543, Adjusted R-squared: 0.5074
F-statistic: 11.82 on 2 and 19 DF, p-value: 0.000463
If “basic” Ho true, MSregression and MSresidual estimate variance and their ratio (F-ratio) = 1
If “basic” Ho false (at least one β ≠ 0) MSregression estimates variance + partial regression slope and their ratio (F-ratio) will be > 1 - F-ratio compared to F-distribution for p-value
Lecture 10: Hypotheses
Also: is any specific β = 0 (explanatory role)?
E.g., does LAT have effect on Ant species?
These H’s tested through model comparison
Model 1 - model with 2 predictors X1, X2:
yi= β0 +β1xi1+β2xi2+ εi
Model 2 - To test Ho that β2 = 0 compare fit of model 1 to model 2 with 1 predictor:
yi= β0 +β1xi1+ εi
Lecture 10: Hypotheses
If SSregression of model1=model2, cannot reject Ho β1 = 0
adding X2 did not explain any additional variance
If SSregression of mod1 > mod2, evidence to reject Ho β1 = 0
adding X2 explains more variance
evidence to reject H0: β2 = 2
SS for β1 is SSextraβ1 = Full SSregression - Reduced SSregression
This quantifies how much additional variance X2 explains
Use partial F-test to test Ho β1 = 0 :
\[F_{w,n-p} = \frac{MS_{Extra}}{FULL\ MS_{Residual}} \]Can also use t-test (R provides this value)
Lecture 10: Explained variance
Explained variance (r2) is calculated the same way as for simple regression:
This leads to “Curvature” of the regression (hyper)plane
Lecture 10: Analyses
Interaction terms lead to “Curvature” of the regression (hyper)plane
Lecture 10: Analyses
Adding interactions:
many more predictors (“parameter proliferation”):
2n is required; 6 parameters= 64 terms; 7 parameters= 128
interpretation more complex
When to include interactions? When they make biological sense!!!
Lecture 10: Dummy variables
Multiple Linear Regression accommodates continuous and categorical variables (gender, vegetation type, etc.) Categorical vars as “dummy vars”, n of dummy variables = n-1 categories
Sex M/F:
Need 1 dummy var with two values (0, 1)
Fertility L/M/H:
Need 2 dummy var, each with two values (0, 1): fert1 (0 if L or H, 1 if M), fert2 (1 if H, 0 if L or M)
Fertility
fert1
fert2
Low
0
0
Med
1
0
High
0
1
Lecture 10: Analyses
Coefficients interpreted relative to reference condition
R codes dummy variables automatically
picks “reference” level alphabetically
Dummy variables with more than 2 levels add extra predictor variables to model
Fertility
fert1
fert2
Low
0
0
Med
1
0
High
0
1
Lecture 10: Analyses
Lecture 10: Comparing models
When have multiple predictors (and interactions!)
how to choose “best” model?
Which predictors to include?
Occam’s razor: “best” model balances complexity with fit to data
To chose:
compare “nested” models
Overfitting
getting high r2 just by having more (useless predictors)
so r2 is not a good way of choosing between nested models
Lecture 10: Comparing models
Need to account for increase in fit with added predictors:
Adjusted r2
Akaike’s information criterion (AIC)
Both “penalize” models for extra predictors
Higher adjusted r2 and lower AIC are better when comparing models
