Introduction

Workshop description

This is an intermediate/advanced R course
Appropriate for those with basic knowledge of R
This is not a statistics course!
Learning objectives:
- Learn the R formula interface
- Specify factor contrasts to test specific hypotheses
- Perform model comparisons
- Run and interpret variety of regression models in R

Materials and Setup

Everyone:

Download materials from http://tutorials.iq.harvard.edu/R/Rstatistics.zip
Extract materials from RStatistics.zip (on lab machines right-click -> WinZip -> Extract to here) and move to your desktop.

Launch RStudio

Open the RStudio program from the Windows start menu
Create a project in the Rstatistics folder you downloaded earlier:
- File => New Project => Existing Directory => Browse and select the Rstatistics folder.You can call Jorim or Asuke for more explanation.

Set working directory

It is often helpful to start your R session by setting your working directory so you don’t have to type the full path names to your data and other files

  # set the working directory
  # setwd("~/Desktop/Rstatistics")
  # setwd("C:/Users/dataclass/Desktop/Rstatistics")

You might also start by listing the files in your working directory

  getwd() # where am I?

## [1] "C:/Users/bopiyo/Desktop/codes/Rstatistics/Rstatistics"

  list.files("dataSets") # files in the dataSets folder

## [1] "Exam.rds"          "NatHealth2008MI"   "NatHealth2011.rds"
## [4] "states.dta"        "states.rds"

Load the states data

The states.dta data comes from http://anawida.de/teach/SS14/anawida/4.linReg/data/states.dta.txt and appears to have originally appeared in Statistics with Stata by Lawrence C. Hamilton.

  # read the states data
  states.data <- readRDS("dataSets/states.rds") 
  #get labels
  states.info <- data.frame(attributes(states.data)[c("names", "var.labels")])
  #look at last few labels
  tail(states.info, 8)

##      names                      var.labels
## 14    csat        Mean composite SAT score
## 15    vsat           Mean verbal SAT score
## 16    msat             Mean math SAT score
## 17 percent       % HS graduates taking SAT
## 18 expense Per pupil expenditures prim&sec
## 19  income Median household income, $1,000
## 20    high             % adults HS diploma
## 21 college         % adults college degree

Linear regression

Examine the data before fitting models

Start by examining the data to check for problems.

  # summary of expense and csat columns, all rows
  sts.ex.sat <- subset(states.data, select = c("expense", "csat"))
  summary(sts.ex.sat)

##     expense          csat       
##  Min.   :2960   Min.   : 832.0  
##  1st Qu.:4352   1st Qu.: 888.0  
##  Median :5000   Median : 926.0  
##  Mean   :5236   Mean   : 944.1  
##  3rd Qu.:5794   3rd Qu.: 997.0  
##  Max.   :9259   Max.   :1093.0

  # correlation between expense and csat
  cor(sts.ex.sat)

##            expense       csat
## expense  1.0000000 -0.4662978
## csat    -0.4662978  1.0000000

Plot the data before fitting models

Plot the data to look for multivariate outliers, non-linear relationships etc.

  # scatter plot of expense vs csat
  plot(sts.ex.sat)

Linear regression example

Linear regression models can be fit with the lm() function
For example, we can use lm to predict SAT scores based on per-pupal expenditures:

  # Fit our regression model
  sat.mod <- lm(csat ~ expense, # regression formula
                data=states.data) # data set
  # Summarize and print the results
  summary(sat.mod) # show regression coefficients table

## 
## Call:
## lm(formula = csat ~ expense, data = states.data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -131.811  -38.085    5.607   37.852  136.495 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.061e+03  3.270e+01   32.44  < 2e-16 ***
## expense     -2.228e-02  6.037e-03   -3.69 0.000563 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 59.81 on 49 degrees of freedom
## Multiple R-squared:  0.2174, Adjusted R-squared:  0.2015 
## F-statistic: 13.61 on 1 and 49 DF,  p-value: 0.0005631

Why is the association between expense and SAT scores negative?

Many people find it surprising that the per-capita expenditure on students is negatively related to SAT scores. The beauty of multiple regression is that we can try to pull these apart. What would the association between expense and SAT scores be if there were no difference among the states in the percentage of students taking the SAT?

  summary(lm(csat ~ expense + percent, data = states.data))

## 
## Call:
## lm(formula = csat ~ expense + percent, data = states.data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -62.921 -24.318   1.741  15.502  75.623 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 989.807403  18.395770  53.806  < 2e-16 ***
## expense       0.008604   0.004204   2.046   0.0462 *  
## percent      -2.537700   0.224912 -11.283 4.21e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 31.62 on 48 degrees of freedom
## Multiple R-squared:  0.7857, Adjusted R-squared:  0.7768 
## F-statistic: 88.01 on 2 and 48 DF,  p-value: < 2.2e-16

The lm class and methods

OK, we fit our model. Now what?

Examine the model object:

  class(sat.mod)

## [1] "lm"

  names(sat.mod)

##  [1] "coefficients"  "residuals"     "effects"       "rank"         
##  [5] "fitted.values" "assign"        "qr"            "df.residual"  
##  [9] "xlevels"       "call"          "terms"         "model"

  methods(class = class(sat.mod))[1:9]

## [1] "add1.lm"                   "alias.lm"                 
## [3] "anova.lm"                  "case.names.lm"            
## [5] "coerce,oldClass,S3-method" "confint.lm"               
## [7] "cooks.distance.lm"         "deviance.lm"              
## [9] "dfbeta.lm"

Use function methods to get more information about the fit

  confint(sat.mod)

##                    2.5 %        97.5 %
## (Intercept) 995.01753164 1126.44735626
## expense      -0.03440768   -0.01014361

  # hist(residuals(sat.mod))

Linear Regression Assumptions

Ordinary least squares regression relies on several assumptions, including that the residuals are normally distributed and homoscedastic, the errors are independent and the relationships are linear.
Investigate these assumptions visually by plotting your model:

  par(mar = c(4, 4, 2, 2), mfrow = c(1, 2)) #optional
  plot(sat.mod, which = c(1, 2)) # "which" argument optional

Comparing models

Do congressional voting patterns predict SAT scores over and above expense? Fit two models and compare them:

  # fit another model, adding house and senate as predictors
  sat.voting.mod <-  lm(csat ~ expense + house + senate,
                        data = na.omit(states.data))
  sat.mod <- update(sat.mod, data=na.omit(states.data))
  # compare using the anova() function
  anova(sat.mod, sat.voting.mod)

## Analysis of Variance Table
## 
## Model 1: csat ~ expense
## Model 2: csat ~ expense + house + senate
##   Res.Df    RSS Df Sum of Sq      F  Pr(>F)  
## 1     46 169050                              
## 2     44 149284  2     19766 2.9128 0.06486 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

  coef(summary(sat.voting.mod))

##                  Estimate   Std. Error    t value     Pr(>|t|)
## (Intercept) 1082.93438041 38.633812740 28.0307405 1.067795e-29
## expense       -0.01870832  0.009691494 -1.9303852 6.001998e-02
## house         -1.44243754  0.600478382 -2.4021473 2.058666e-02
## senate         0.49817861  0.513561356  0.9700469 3.373256e-01

Exercise 0: least squares regression

Use the states.rds data set. Fit a model predicting energy consumed per capita (energy) from the percentage of residents living in metropolitan areas (metro). Be sure to

Examine/plot the data before fitting the model
Print and interpret the model summary
plot the model to look for deviations from modeling assumptions

Select one or more additional predictors to add to your model and repeat steps 1-3. Is this model significantly better than the model with metro as the only predictor?

Interactions and factors

Modeling interactions

Interactions allow us assess the extent to which the association between one predictor and the outcome depends on a second predictor. For example: Does the association between expense and SAT scores depend on the median income in the state?

    #Add the interaction to the model
  sat.expense.by.percent <- lm(csat ~ expense*income,
                               data=states.data) 
  #Show the results
    coef(summary(sat.expense.by.percent)) # show regression coefficients table

##                     Estimate   Std. Error   t value     Pr(>|t|)
## (Intercept)     1.380364e+03 1.720863e+02  8.021351 2.367069e-10
## expense        -6.384067e-02 3.270087e-02 -1.952262 5.687837e-02
## income         -1.049785e+01 4.991463e+00 -2.103161 4.083253e-02
## expense:income  1.384647e-03 8.635529e-04  1.603431 1.155395e-01

Regression with categorical predictors

Let’s try to predict SAT scores from region, a categorical variable. Note that you must make sure R does not think your categorical variable is numeric.

  # make sure R knows region is categorical
  str(states.data$region)

##  Factor w/ 4 levels "West","N. East",..: 3 1 1 3 1 1 2 3 NA 3 ...

  states.data$region <- factor(states.data$region)
  #Add region to the model
  sat.region <- lm(csat ~ region,
                   data=states.data) 
  #Show the results
  coef(summary(sat.region)) # show regression coefficients table

##                Estimate Std. Error    t value     Pr(>|t|)
## (Intercept)   946.30769   14.79582 63.9577807 1.352577e-46
## regionN. East -56.75214   23.13285 -2.4533141 1.800383e-02
## regionSouth   -16.30769   19.91948 -0.8186806 4.171898e-01
## regionMidwest  63.77564   21.35592  2.9863209 4.514152e-03

  anova(sat.region) # show ANOVA table

## Analysis of Variance Table
## 
## Response: csat
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## region     3  82049 27349.8  9.6102 4.859e-05 ***
## Residuals 46 130912  2845.9                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Again, make sure to tell R which variables are categorical by converting them to factors!

Setting factor reference groups and contrasts

In the previous example we use the default contrasts for region. The default in R is treatment contrasts, with the first level as the reference. We can change the reference group or use another coding scheme using the C function.

  # print default contrasts
  contrasts(states.data$region)

##         N. East South Midwest
## West          0     0       0
## N. East       1     0       0
## South         0     1       0
## Midwest       0     0       1

  # change the reference group
  coef(summary(lm(csat ~ C(region, base=4),
                  data=states.data)))

##                        Estimate Std. Error   t value     Pr(>|t|)
## (Intercept)          1010.08333   15.39998 65.589930 4.296307e-47
## C(region, base = 4)1  -63.77564   21.35592 -2.986321 4.514152e-03
## C(region, base = 4)2 -120.52778   23.52385 -5.123641 5.798399e-06
## C(region, base = 4)3  -80.08333   20.37225 -3.931000 2.826007e-04

  # change the coding scheme
  coef(summary(lm(csat ~ C(region, contr.helmert),
                  data=states.data)))

##                             Estimate Std. Error     t value     Pr(>|t|)
## (Intercept)               943.986645   7.706155 122.4977451 1.689670e-59
## C(region, contr.helmert)1 -28.376068  11.566423  -2.4533141 1.800383e-02
## C(region, contr.helmert)2   4.022792   5.884552   0.6836191 4.976450e-01
## C(region, contr.helmert)3  22.032229   4.446777   4.9546509 1.023364e-05

See also ?contrasts, ?contr.treatment, and ?relevel.

Exercise 1: interactions and factors

Use the states data set.

Add on to the regression equation that you created in exercise 1 by generating an interaction term and testing the interaction.
Try adding region to the model. Are there significant differences across the four regions?

Regression with binary outcomes

Logistic regression

This far we have used the lm function to fit our regression models. lm is great, but limited–in particular it only fits models for continuous dependent variables. For categorical dependent variables we can use the glm() function.

For these models we will use a different dataset, drawn from the National Health Interview Survey. From the CDC website:

‘> The National Health Interview Survey (NHIS) has monitored the health of the nation since 1957. NHIS data on a broad range of health topics are collected through personal household interviews. For over 50 years, the U.S. Census Bureau has been the data collection agent for the National Health Interview Survey. Survey results have been instrumental in providing data to track health status, health care access, and progress toward achieving national health objectives.’

Load the National Health Interview Survey data:

  NH11 <- readRDS("dataSets/NatHealth2011.rds")
  labs <- attributes(NH11)$labels

Logistic regression example

Let’s predict the probability of being diagnosed with hypertension based on age, sex, sleep, and bmi

  str(NH11$hypev) # check stucture of hypev

##  Factor w/ 5 levels "1 Yes","2 No",..: 2 2 1 2 2 1 2 2 1 2 ...

  levels(NH11$hypev) # check levels of hypev

## [1] "1 Yes"             "2 No"              "7 Refused"        
## [4] "8 Not ascertained" "9 Don't know"

  # collapse all missing values to NA
  NH11$hypev <- factor(NH11$hypev, levels=c("2 No", "1 Yes"))
  # run our regression model
  hyp.out <- glm(hypev~age_p+sex+sleep+bmi,
                data=NH11, family="binomial")
  coef(summary(hyp.out))

##                 Estimate   Std. Error    z value     Pr(>|z|)
## (Intercept) -4.269466028 0.0564947294 -75.572820 0.000000e+00
## age_p        0.060699303 0.0008227207  73.778743 0.000000e+00
## sex2 Female -0.144025092 0.0267976605  -5.374540 7.677854e-08
## sleep       -0.007035776 0.0016397197  -4.290841 1.779981e-05
## bmi          0.018571704 0.0009510828  19.526906 6.485172e-85

Logistic regression coefficients

Generalized linear models use link functions, so raw coefficients are difficult to interpret. For example, the age coefficient of .06 in the previous model tells us that for every one unit increase in age, the log odds of hypertension diagnosis increases by 0.06. Since most of us are not used to thinking in log odds this is not too helpful!

One solution is to transform the coefficients to make them easier to interpret

  hyp.out.tab <- coef(summary(hyp.out))
  hyp.out.tab[, "Estimate"] <- exp(coef(hyp.out))
  hyp.out.tab

##               Estimate   Std. Error    z value     Pr(>|z|)
## (Intercept) 0.01398925 0.0564947294 -75.572820 0.000000e+00
## age_p       1.06257935 0.0008227207  73.778743 0.000000e+00
## sex2 Female 0.86586602 0.0267976605  -5.374540 7.677854e-08
## sleep       0.99298892 0.0016397197  -4.290841 1.779981e-05
## bmi         1.01874523 0.0009510828  19.526906 6.485172e-85

Generating predicted values

In addition to transforming the log-odds produced by glm to odds, we can use the predict() function to make direct statements about the predictors in our model. For example, we can ask “How much more likely is a 63 year old female to have hypertension compared to a 33 year old female?”.

  # Create a dataset with predictors set at desired levels
  predDat <- with(NH11,
                  expand.grid(age_p = c(33, 63),
                              sex = "2 Female",
                              bmi = mean(bmi, na.rm = TRUE),
                              sleep = mean(sleep, na.rm = TRUE)))
  # predict hypertension at those levels
  cbind(predDat, predict(hyp.out, type = "response",
                         se.fit = TRUE, interval="confidence",
                         newdata = predDat))

##   age_p      sex      bmi   sleep       fit      se.fit residual.scale
## 1    33 2 Female 29.89565 7.86221 0.1289227 0.002849622              1
## 2    63 2 Female 29.89565 7.86221 0.4776303 0.004816059              1

This tells us that a 33 year old female has a 13% probability of having been diagnosed with hypertension, while and 63 year old female has a 48% probability of having been diagnosed.

Packages for computing and graphing predicted values

Instead of doing all this ourselves, we can use the effects package to compute quantities of interest for us (cf. the Zelig package).

  library(effects)

## Loading required package: carData

## lattice theme set by effectsTheme()
## See ?effectsTheme for details.

  plot(allEffects(hyp.out))

Exercise 2: logistic regression

Use the NH11 data set that we loaded earlier.

Use glm to conduct a logistic regression to predict ever worked (everwrk) using age (age_p) and marital status (r_maritl).
Predict the probability of working for each level of marital status.

Note that the data is not perfectly clean and ready to be modeled. You will need to clean up at least some of the variables before fitting the model.

Multilevel Modeling

Multilevel modeling overview

Multi-level (AKA hierarchical) models are a type of mixed-effects models
Used to model variation due to group membership where the goal is to generalize to a population of groups
Can model different intercepts and/or slopes for each group
Mixed-effecs models include two types of predictors: fixed-effects and random effects
- Fixed-effects – observed levels are of direct interest (.e.g, sex, political party…)
- Random-effects – observed levels not of direct interest: goal is to make inferences to a population represented by observed levels
- In R the lme4 package is the most popular for mixed effects models
- Use the lmer function for liner mixed models, glmer for generalized mixed models

  library(lme4)

## Loading required package: Matrix

The Exam data

The Exam data set contans exam scores of 4,059 students from 65 schools in Inner London. The variable names are as follows:

variable	Description
school	School ID - a factor.
normexam	Normalized exam score.
schgend	School gender - a factor. Levels are ‘mixed’, ‘boys’, and ‘girls’.
schavg	School average of intake score.
vr	Student level Verbal Reasoning (VR) score band at intake - ‘bottom 25%’, ‘mid 50%’, and ‘top 25%’.
intake	Band of student’s intake score - a factor. Levels are ‘bottom 25%’, ‘mid 50%’ and ‘top 25%’./
standLRT	Standardised LR test score.
sex	Sex of the student - levels are ‘F’ and ‘M’.
type	School type - levels are ‘Mxd’ and ‘Sngl’.
student	Student id (within school) - a factor

  Exam <- readRDS("dataSets/Exam.rds")

The null model and ICC

As a preliminary step it is often useful to partition the variance in the dependent variable into the various levels. This can be accomplished by running a null model (i.e., a model with a random effects grouping structure, but no fixed-effects predictors).

  # null model, grouping by school but not fixed effects.
  Norm1 <-lmer(normexam ~ 1 + (1|school),
                data=na.omit(Exam), REML = FALSE)
  summary(Norm1)

## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: normexam ~ 1 + (1 | school)
##    Data: na.omit(Exam)
## 
##      AIC      BIC   logLik deviance df.resid 
##   9964.9   9983.5  -4979.4   9958.9     3659 
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.8751 -0.6460  0.0045  0.6898  3.6842 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  school   (Intercept) 0.1606   0.4007  
##  Residual             0.8520   0.9231  
## Number of obs: 3662, groups:  school, 65
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept) -0.01301    0.05270  -0.247

The is .169/(.169 + .848) = .17: 17% of the variance is at the school level.

Adding fixed-effects predictors

Predict exam scores from student’s standardized tests scores

  Norm2 <-lmer(normexam~standLRT + (1|school),
               data=na.omit(Exam),
               REML = FALSE) 
  summary(Norm2)

## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: normexam ~ standLRT + (1 | school)
##    Data: na.omit(Exam)
## 
##      AIC      BIC   logLik deviance df.resid 
##   8480.1   8505.0  -4236.1   8472.1     3658 
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.6722 -0.6300  0.0236  0.6770  3.3347 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  school   (Intercept) 0.09197  0.3033  
##  Residual             0.56909  0.7544  
## Number of obs: 3662, groups:  school, 65
## 
## Fixed effects:
##              Estimate Std. Error t value
## (Intercept) 1.913e-05  4.022e-02    0.00
## standLRT    5.670e-01  1.324e-02   42.84
## 
## Correlation of Fixed Effects:
##          (Intr)
## standLRT 0.007

Multiple degree of freedom comparisons

As with lm and glm models, you can compare the two lmer models using the anova function.

  anova(Norm1, Norm2)

## Data: na.omit(Exam)
## Models:
## Norm1: normexam ~ 1 + (1 | school)
## Norm2: normexam ~ standLRT + (1 | school)
##       Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)    
## Norm1  3 9964.9 9983.5 -4979.4   9958.9                             
## Norm2  4 8480.1 8505.0 -4236.1   8472.1 1486.8      1  < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Random slopes

Add a random effect of students’ standardized test scores as well. Now in addition to estimating the distribution of intercepts across schools, we also estimate the distribution of the slope of exam on standardized test.

  Norm3 <- lmer(normexam~standLRT + (standLRT|school), 
                data = na.omit(Exam),
                REML = FALSE) 
  summary(Norm3)

## Linear mixed model fit by maximum likelihood  ['lmerMod']
## Formula: normexam ~ standLRT + (standLRT | school)
##    Data: na.omit(Exam)
## 
##      AIC      BIC   logLik deviance df.resid 
##   8444.1   8481.4  -4216.1   8432.1     3656 
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -3.7894 -0.6233  0.0242  0.6736  3.4372 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev. Corr
##  school   (Intercept) 0.08926  0.2988       
##           standLRT    0.01592  0.1262   0.50
##  Residual             0.55591  0.7456       
## Number of obs: 3662, groups:  school, 65
## 
## Fixed effects:
##             Estimate Std. Error t value
## (Intercept) -0.01237    0.03978  -0.311
## standLRT     0.56111    0.02100  26.714
## 
## Correlation of Fixed Effects:
##          (Intr)
## standLRT 0.361

Test the significance of the random slope

To test the significance of a random slope just compare models with and without the random slope term

  anova(Norm2, Norm3)

## Data: na.omit(Exam)
## Models:
## Norm2: normexam ~ standLRT + (1 | school)
## Norm3: normexam ~ standLRT + (standLRT | school)
##       Df    AIC    BIC  logLik deviance Chisq Chi Df Pr(>Chisq)    
## Norm2  4 8480.1 8505.0 -4236.1   8472.1                            
## Norm3  6 8444.1 8481.4 -4216.1   8432.1 40.01      2  2.051e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

–>

<!– –>

Wrap-up

-this compilation was made possible by Benard Asuke and Jorim Opiyo -For any query or compliments kindly contact us

R regression models workshop notes compiled by Benard Asuke and Jorim Opiyo Ayugi-This will reduce your boredom during this quarantine