R Bootcamp Day 3 Part 1 - Statistics in R
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1. The document provides an overview of common statistical tests and functions in R, including linear regression and mixed models. It demonstrates how to perform t-tests, linear regression, and fit mixed models using the lm() and lme() functions. 2. Linear regression in R uses Wilkinson-Rogers notation to specify models in the lm() function. Mixed models can include both fixed and random effects, as demonstrated by fitting a basic mixed model with a random intercept to a subset of the Orthodont data. 3. Class lm objects have their own plot() function to visualize linear regression results. Summary() provides statistics on model fits while individual fields like coefficients and residuals can be accessed directly from the model object.
![R: Linear regression
• We can also look at individual fields of the lm object.
car.fit$coefficients
(Intercept) speed
-17.579095 3.932409
car.fit$residuals[1:3]
1 2 3
3.849460 11.849460 -5.947766
car.fit$fitted.values[1:3]
1 2 3
-1.849460 -1.849460 9.947766](https://image.slidesharecdn.com/davisboot3-170511142037/85/R-Bootcamp-Day-3-Part-1-Statistics-in-R-16-320.jpg)
![R: Mixed models
OrthoFem <-
Orthodont[Orthodont$Sex
== "Female", ]
plot(OrthoFem)](https://image.slidesharecdn.com/davisboot3-170511142037/85/R-Bootcamp-Day-3-Part-1-Statistics-in-R-24-320.jpg)
R Bootcamp Day 3 Part 1 - Statistics in R
- 1. R Bootcamp Day 3 Part 1 Jefferson Davis Olga Scrivner
- 2. Day 2 stuff From yesterday and the day before • R values have types/classes such as numeric, character, logical, dataframes, and matrices. • Much of R functionality is in libraries • For help on a function run ? t.test() from the R console. • The plot() function will usually do something useful.
- 3. R: Common stats functions Common statistical tests are very straightforward in R. Let's try one on yesterday's dataset cars of car speeds and stopping distances from the 1920s. head(cars) speed dist 1 4 2 2 4 10 3 7 4 4 7 22 5 8 16 6 9 10
- 4. R: Common stats functions Here's a t-test that the mean of the speeds in cars is not 12. t.test(cars$speed, mu=12) One Sample t-test data: cars$speed t = 4.5468, df = 49, p-value = 3.588e-05 alternative hypothesis: true mean is not equal to 12 95 percent confidence interval: 13.89727 16.90273 sample estimates: mean of x 15.4
- 5. R: Common stats functions We can change the parameters of t-test. t.test(cars$speed, mu=12, alternative="less", conf.level=.99) One Sample t-test data: cars$speed t = 4.5468, df = 49, p-value = 1 alternative hypothesis: true mean is less than 12 99 percent confidence interval: -Inf 17.19834 sample estimates: mean of x 15.4
- 6. R: Common stats functions Anything you would see in a year long stats sequence will have an implentation in R. chisq.test() #Chi-squared prop.test() #Proportions test binom.test() #Exact binomial test ks.test() #Kolmogorov–Smirnov sd() #Standard deviation cor() #Correlation
- 7. R: Linear regression Regression analysis is one of the most popular and important tools in statistics. If R goofed here, it would be worthless. R uses the function lm() for linear models. The regression formula is given in Wilkinson-Rogers notation Predictor terms Wilkinson Notation Intercept 1 (Default) No intercept -1 x1 x1 x1, x2 x1 + x2 x1, x2, x1x2 x1*x2 (or x1 + x2 + x1:x2) x1x2 x1:x2 x1 2, x1 x1^2 x1 + x2 I(x1 + x2) (The letter I)
- 8. R: Linear regression Regression analysis is one of the most important tools in statistics. R uses Wilkinson-Rogers notation to to specify linear models. So a model such as yi = β0 + β1 xi1 + εi Shows up in the R syntax as y ~ x1 Let's review this syntax. (Tables from https://www.mathworks.com/help/stats/wilkinson- notation.html)
- 9. R: Linear regression Predictor terms Wilkinson Notation Intercept 1 (Default) No intercept -1 x1 x1 x1, x2 x1 + x2 x1, x2, x1x2 x1*x2 (or x1 + x2 + x1:x2) x1x2 x1:x2 x1 2, x1 x1^2 x1 + x2 I(x1 + x2)
- 10. R: Linear regression Model Wilkinson Notation yi = β0 + β1 xi1 + β2 xi2 + εi Two predictors y ~ x1 + x2 yi = β1 xi1 + β2 xi2 + εi Two predictors and no intercept y ~ x1 + x2 - 1 yi = β0 + β1 xi1 + β2 xi2 + β3 xi1 xi2 + εi Two predictors with the interaction term y ~ x1 * x2 y ~ x1 + x2 + x1:x2 yi = β0 + β1 (xi1 + xi2 ) + εi Regressing on the sum of predictors y ~ I(x1 + x2) yi = β0 + β1 xi1 + β2 xi2 + β3 xi3 + β4 xi1 xi2 + εi Three predictors with one interaction y ~ x1 * x2 + x3
- 11. R: Linear regression Model terms Wilkinson Notation yi = β1 xi1 + β2 xi2 + β3 xi1 xi2 + εi Two predictors, no intercept yi = β0 + β1 xi1 + β2 xi2 + β3 xi3 + β4 xi1 xi2 + β5 xi1 xi3 + β6 xi2 xi3 + β7 xi1 xi2xi3+ εi Three predictors, all interaction terms yi = β0 + β1 xi1 + β2 xi2 + β3 xi3 + β4 xi1 xi2 + β5 xi1 xi3 + β6 xi2 xi3 + εi Three predictors, all two-way interaction terms.
- 12. R: Linear regression Model terms Wilkinson Notation yi = β1 xi1 + β2 xi2 + β3 xi1 xi2 + εi Two predictors, no intercept y ~ x1*x2 - 1 yi = β0 + β1 xi1 + β2 xi2 + β3 xi3 + β4 xi1 xi2 + β5 xi1 xi3 + β6 xi2 xi3 + β7 xi1 xi2xi3+ εi Three predictors, all interaction terms y ~ x1 * x2 * x3 yi = β0 + β1 xi1 + β2 xi2 + β3 xi3 + β4 xi1 xi2 + β5 xi1 xi3 + β6 xi2 xi3 + εi Three predictors, all two-way interaction terms y ~ x1 * x2 * x3 – x1:x2:x3
- 13. R: Linear regression • R uses the function lm() for linear models. • Generic syntax lm(DV ~ IV1, NAME_OF_DATAFRAME) • The above tells R that to regress the dependent variable (DV) onto independent variable IV1. We can include other variables and interaction effects. lm(DV ~ IV1 + IV2 + IV1*IV2, NAME_OF_DATAFRAME)
- 14. R: Linear regression • Let's do an example using the cars data set. How about regressing stopping distance on speed. lm(dist ~ speed, cars) Call:lm(formula = dist ~ speed, data = cars) Coefficients: (Intercept) speed -17.579 3.932 • To work more let's store this in a variable car.fit <- lm(dist ~ speed, cars)
- 15. R: Linear regression summary(car.fit) Call: lm(formula = dist ~ speed, data = cars) Residuals: Min 1Q Median 3Q Max -29.069 -9.525 -2.272 9.215 43.201 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -17.5791 6.7584 -2.601 0.0123 * speed 3.9324 0.4155 9.464 1.49e-12 ***
- 16. R: Linear regression • We can also look at individual fields of the lm object. car.fit$coefficients (Intercept) speed -17.579095 3.932409 car.fit$residuals[1:3] 1 2 3 3.849460 11.849460 -5.947766 car.fit$fitted.values[1:3] 1 2 3 -1.849460 -1.849460 9.947766
- 17. R: Linear regression • Plot the fit plot(cars$speed, cars$dist, xlab = "distance", ylab = "speed") abline(car.fit, col="red")
- 18. R: Linear regression • Class lm object have their own overloaded plot() function plot(car.fit)
- 19. R: Linear regression • Class lm object have their own overloaded plot() function plot(car.fit)
- 20. R: Linear regression • Class lm object have their own overloaded plot() function plot(car.fit)
- 21. R: Mixed models It doesn't seem crazy to fit a slope but use a random effect for intercept. fmOrthF <- lme( distance ~ age, data = OrthoFem, random = ~ 1 | Subject )
- 22. R: Linear regression • Class lm object have their own overloaded plot() function plot(car.fit)
- 23. R: Mixed models • Let's take a look at a mixed model. We need a more complex dataset. We use a subset of the Orthodont data set from the Nonlinear Mixed-Effects Models (nlme) library. library(nlme) head(Orthodont) Grouped Data: distance ~ age | Subject distance age Subject Sex 1 26.0 8 M01 Male 2 25.0 10 M01 Male 3 29.0 12 M01 Male 4 31.0 14 M01 Male
- 24. R: Mixed models OrthoFem <- Orthodont[Orthodont$Sex == "Female", ] plot(OrthoFem)
- 25. R: Mixed models In fact, it isn't crazy. summary(fmOrthF) Linear mixed-effects model fit by REML Data: OrthoFem AIC BIC logLik 149.2183 156.169 -70.60916 Random effects: Formula: ~1 | Subject (Intercept) Residual StdDev: 2.06847 0.7800331 Fixed effects: distance ~ age Value Std.Error DF t-value p-value (Intercept) 17.372727 0.8587419 32 20.230440 0 age 0.479545 0.0525898 32 9.118598 0 Correlation: (Intr)age -0.674
- 26. R: Conditional trees At this point, I tag Olga in.