PROBLEM SET 4
Bienuel Esmeralda
2023-12-18
\[ INTRODUCTION \space TO \space REGRESSION \space IN \space R \]
Simple Linear Regression
Regression analysis is a statistical tool that is used to explain relationships between outcome variables as a function of one or more predictor variables. A linear regression model, we assume that the relationship can be explained with a linear function, in a simple linear regression model, it only has one outcome variable and one predictor variable.
Linear Function
The data in an SRM are in pairs. usually, in high school statistics, correlation topic, x will be the independent variable, and y is the dependent variable, same here also.
\(X:\) The predictor, \(x_1,x_2,…,x_n\) where n are observations from \(X\)
\(Y\): The response, \(y_1,y_2,..,y_n\) where n are observations from \(Y\)
A linear function can be determined by the intercept of and the slope of the line in this model,
\[ y=\beta_1x+\beta_0 \tag{1.1} \]
where \(\beta_1\) is the slope and \(\beta_0\) is the intercept.
In plotting the model in a Cartesian plane, we will use the method rise over run, \(\frac{\Delta y}{\Delta x}\).
Linear Regression as a statistical model
The model in (1.1) is not really applicable in real life situations because errors will occur in an obtained data. The regression model has two parts, linear function and an error term. \(\beta_0\) and \(\beta_1\) are called parameters, and are estimated from the data.
\[ Y=\beta_0 +\beta_1x+\epsilon \tag{1.2} \]
The linear function predicts the mean value of the outcome for a given value of the predictor variable.
The error term is the unexplained uncertainty of the model and is a random variable with mean equal to zero and unknown variance \(\sigma^2\). The error usually, assumed to have a normal distribution. In a simple regression model, for each individual or unit \(i=1,2,..,n\) in the study, we have a pair of observations, \((x_i,y_i)\).
Ordinary Least Squares
Decomposing the sum of squares and ANOVA
Running the regression model in RStudio
d <- read.csv("C:/Users/USER/Dropbox/PC/Downloads/elemapi2v2.csv")The data is in .csv format, and we use the read.csv function to read and load it in R. The resulting output will be a data.frame object that will appear in the environment tab. By clicking on the data.frame object in the environment pane we can view it as a spreadsheet.
Checking the nature of the data to make sure it is applicable for Linear regression.
Using the function class. , it will determine what is the environment is your data is.
class(d)## [1] "data.frame"Using the function names. , it will return the names of the variables which is set in column.
names(d)## [1] "snum" "dnum" "api00" "api99" "growth" "meals"
## [7] "ell" "yr_rnd" "mobility" "acs_k3" "acs_46" "not_hsg"
## [13] "hsg" "some_col" "col_grad" "grad_sch" "avg_ed" "full"
## [19] "emer" "enroll" "mealcat" "collcat" "abv_hsg" "lgenroll"Using the dim. , function determines how many is your row (observations) and your column (variables).
dim(d)## [1] 400 24d## snum dnum api00 api99 growth meals ell yr_rnd mobility acs_k3 acs_46
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## not_hsg hsg some_col col_grad grad_sch avg_ed full emer enroll mealcat
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## 202 1 11 2.847573
## 203 1 67 2.372912
## 204 1 61 2.698101
## 205 1 72 2.846955
## 206 2 59 2.868056
## 207 2 94 2.822822
## 208 3 96 2.705864
## 209 2 98 2.550228
## 210 1 63 3.195900
## 211 1 98 2.492760
## 212 2 100 2.713491
## 213 3 90 2.707570
## 214 3 95 2.625312
## 215 1 100 2.582063
## 216 3 96 2.380211
## 217 3 94 2.584331
## 218 3 93 2.522444
## 219 1 100 2.622214
## 220 2 45 2.597695
## 221 3 93 2.271842
## 222 3 80 2.371068
## 223 1 100 2.718502
## 224 3 70 2.296665
## 225 2 69 2.779596
## 226 2 66 2.615950
## 227 2 88 2.693727
## 228 2 81 2.406540
## 229 2 96 2.252853
## 230 3 69 2.507856
## 231 3 80 2.281033
## 232 2 89 2.774517
## 233 2 49 2.808211
## 234 1 70 2.564666
## 235 3 98 2.292256
## 236 2 100 2.328380
## 237 1 100 2.296665
## 238 1 100 2.522444
## 239 2 99 2.737987
## 240 2 53 2.835056
## 241 3 81 2.494155
## 242 1 99 2.842609
## 243 2 100 2.594393
## 244 2 100 2.664642
## 245 2 100 2.827369
## 246 3 92 2.733197
## 247 2 100 2.782473
## 248 2 90 2.521138
## 249 1 54 2.523746
## 250 2 94 2.694605
## 251 1 98 2.503791
## 252 2 72 2.545307
## 253 3 95 2.502427
## 254 3 75 2.584331
## 255 3 85 2.743510
## 256 3 76 2.758155
## 257 3 76 2.385606
## 258 3 79 2.744293
## 259 3 97 2.790285
## 260 2 61 2.733197
## 261 3 86 2.711807
## 262 3 93 2.690196
## 263 3 100 2.315970
## 264 3 78 2.817565
## 265 2 69 2.809560
## 266 2 63 2.637490
## 267 3 96 2.489958
## 268 1 100 2.346353
## 269 3 100 2.603144
## 270 2 90 2.517196
## 271 3 72 2.361728
## 272 1 25 2.230449
## 273 3 81 2.553883
## 274 2 92 2.123852
## 275 2 100 2.608526
## 276 2 84 2.465383
## 277 2 85 2.748188
## 278 3 94 2.612784
## 279 3 96 2.767898
## 280 2 84 2.731589
## 281 2 92 2.307496
## 282 2 69 2.715167
## 283 2 97 2.707570
## 284 2 47 2.770852
## 285 2 70 2.717671
## 286 2 81 2.765669
## 287 1 45 2.809560
## 288 3 72 2.442480
## 289 3 75 2.783189
## 290 3 77 2.642465
## 291 1 40 2.765669
## 292 2 52 2.672098
## 293 2 39 2.322219
## 294 3 93 2.738781
## 295 3 90 2.130334
## 296 1 81 2.363612
## 297 3 93 2.469822
## 298 3 88 2.809560
## 299 3 79 2.641474
## 300 1 97 2.671173
## 301 3 88 2.759668
## 302 2 56 2.709270
## 303 3 99 2.780317
## 304 3 90 2.113943
## 305 1 45 2.860937
## 306 3 94 2.720159
## 307 3 96 2.536558
## 308 2 55 2.602060
## 309 3 89 2.403121
## 310 3 74 2.549003
## 311 1 55 2.544068
## 312 3 90 2.786751
## 313 3 98 2.734800
## 314 1 99 2.745075
## 315 3 95 2.577492
## 316 2 97 2.408240
## 317 1 30 2.612784
## 318 1 47 2.882525
## 319 3 76 2.201397
## 320 1 53 2.661813
## 321 3 95 2.547775
## 322 2 88 2.334454
## 323 3 91 2.424882
## 324 3 91 2.426511
## 325 2 81 2.294466
## 326 2 98 2.374748
## 327 2 92 2.517196
## 328 2 78 2.525045
## 329 2 69 2.313867
## 330 1 91 2.271842
## 331 2 93 2.447158
## 332 1 99 2.651278
## 333 3 91 2.359835
## 334 2 69 2.332438
## 335 1 53 2.276462
## 336 1 100 2.606381
## 337 1 96 2.346353
## 338 3 100 2.380211
## 339 1 100 2.779596
## 340 1 98 2.587711
## 341 3 95 2.544068
## 342 2 59 2.514548
## 343 2 68 2.409933
## 344 3 94 2.448706
## 345 3 84 2.536558
## 346 1 72 2.491362
## 347 3 99 2.527630
## 348 3 99 2.501059
## 349 3 92 2.472756
## 350 3 70 2.429752
## 351 3 84 2.462398
## 352 2 100 2.550228
## 353 3 93 2.530200
## 354 3 99 2.595496
## 355 3 89 2.477121
## 356 3 90 2.164353
## 357 3 83 2.506505
## 358 3 94 2.564666
## 359 3 90 2.659916
## 360 3 99 2.502427
## 361 3 97 2.600973
## 362 3 67 2.612784
## 363 3 99 2.494155
## 364 3 93 2.346353
## 365 2 57 2.779596
## 366 1 46 2.891537
## 367 1 17 2.770852
## 368 2 54 2.786041
## 369 3 90 2.841985
## 370 3 78 2.837588
## 371 1 42 2.773055
## 372 1 50 2.565848
## 373 1 100 2.654177
## 374 1 50 2.817565
## 375 1 100 2.572872
## 376 1 100 2.598791
## 377 1 100 2.593286
## 378 3 69 2.546543
## 379 1 25 2.728354
## 380 2 67 2.817565
## 381 3 78 2.725912
## 382 2 59 2.786751
## 383 3 82 2.747412
## 384 1 60 2.559907
## 385 1 50 2.477121
## 386 2 73 2.519828
## 387 3 68 2.847573
## 388 3 87 2.668386
## 389 2 79 2.561101
## 390 2 92 2.556303
## 391 2 64 2.817565
## 392 1 48 2.592177
## 393 1 60 2.338456
## 394 2 73 2.602060
## 395 2 93 2.501059
## 396 2 95 2.424882
## 397 3 92 2.663701
## 398 2 97 2.556303
## 399 1 90 2.478566
## 400 1 69 2.429752Exploring the data
Descriptive analysis will help us to understand our data better and investigate if there are any problems in the data. In this workshop we skip this part, but in a real study descriptive analysis is an important part of a good data analysis.
In Descriptive analysis we usually get the Central Tendency and Measurement of variance.
str(d)## 'data.frame': 400 obs. of 24 variables:
## $ snum : int 906 889 887 876 888 4284 4271 2910 2899 2887 ...
## $ dnum : int 41 41 41 41 41 98 98 108 108 108 ...
## $ api00 : int 693 570 546 571 478 858 918 831 860 737 ...
## $ api99 : int 600 501 472 487 425 844 864 791 838 703 ...
## $ growth : int 93 69 74 84 53 14 54 40 22 34 ...
## $ meals : int 67 92 97 90 89 10 5 2 5 29 ...
## $ ell : int 9 21 29 27 30 3 2 3 6 15 ...
## $ yr_rnd : int 0 0 0 0 0 0 0 0 0 0 ...
## $ mobility: int 11 33 36 27 44 10 16 44 10 17 ...
## $ acs_k3 : int 16 15 17 20 18 20 19 20 20 21 ...
## $ acs_46 : int 22 32 25 30 31 33 28 31 30 29 ...
## $ not_hsg : int 0 0 0 36 50 1 1 0 2 8 ...
## $ hsg : int 0 0 0 45 50 8 4 4 9 25 ...
## $ some_col: int 0 0 0 9 0 24 18 16 15 34 ...
## $ col_grad: int 0 0 0 9 0 36 34 50 42 27 ...
## $ grad_sch: int 0 0 0 0 0 31 43 30 33 7 ...
## $ avg_ed : num NA NA NA 1.91 1.5 ...
## $ full : int 76 79 68 87 87 100 100 96 100 96 ...
## $ emer : int 24 19 29 11 13 0 0 2 0 7 ...
## $ enroll : int 247 463 395 418 520 343 303 1513 660 362 ...
## $ mealcat : int 2 3 3 3 3 1 1 1 1 1 ...
## $ collcat : int 1 1 1 1 1 2 2 2 2 3 ...
## $ abv_hsg : int 100 100 100 64 50 99 99 100 98 92 ...
## $ lgenroll: num 2.39 2.67 2.6 2.62 2.72 ...summary(d)## snum dnum api00 api99
## Min. : 58 Min. : 41.0 Min. :369.0 Min. :333.0
## 1st Qu.:1720 1st Qu.:395.0 1st Qu.:523.8 1st Qu.:484.8
## Median :3008 Median :401.0 Median :643.0 Median :602.0
## Mean :2867 Mean :457.7 Mean :647.6 Mean :610.2
## 3rd Qu.:4198 3rd Qu.:630.0 3rd Qu.:762.2 3rd Qu.:731.2
## Max. :6072 Max. :796.0 Max. :940.0 Max. :917.0
##
## growth meals ell yr_rnd
## Min. :-69.00 Min. : 0.00 Min. : 0.00 Min. :0.00
## 1st Qu.: 19.00 1st Qu.: 31.00 1st Qu.: 9.75 1st Qu.:0.00
## Median : 36.00 Median : 67.50 Median :25.00 Median :0.00
## Mean : 37.41 Mean : 60.31 Mean :31.45 Mean :0.23
## 3rd Qu.: 53.25 3rd Qu.: 90.00 3rd Qu.:50.25 3rd Qu.:0.00
## Max. :134.00 Max. :100.00 Max. :91.00 Max. :1.00
##
## mobility acs_k3 acs_46 not_hsg
## Min. : 2.00 Min. :14.00 Min. :20.00 Min. : 0.00
## 1st Qu.:13.00 1st Qu.:18.00 1st Qu.:27.00 1st Qu.: 4.00
## Median :17.00 Median :19.00 Median :29.00 Median : 14.00
## Mean :18.25 Mean :19.16 Mean :29.69 Mean : 21.25
## 3rd Qu.:22.00 3rd Qu.:20.00 3rd Qu.:31.00 3rd Qu.: 34.00
## Max. :47.00 Max. :25.00 Max. :50.00 Max. :100.00
## NA's :1 NA's :2 NA's :3
## hsg some_col col_grad grad_sch
## Min. : 0.00 Min. : 0.00 Min. : 0.0 Min. : 0.000
## 1st Qu.: 17.00 1st Qu.:12.00 1st Qu.: 7.0 1st Qu.: 1.000
## Median : 26.00 Median :19.00 Median : 16.0 Median : 4.000
## Mean : 26.02 Mean :19.71 Mean : 19.7 Mean : 8.637
## 3rd Qu.: 34.00 3rd Qu.:28.00 3rd Qu.: 30.0 3rd Qu.:10.000
## Max. :100.00 Max. :67.00 Max. :100.0 Max. :67.000
##
## avg_ed full emer enroll
## Min. :1.000 Min. : 37.00 Min. : 0.00 Min. : 130.0
## 1st Qu.:2.070 1st Qu.: 76.00 1st Qu.: 3.00 1st Qu.: 320.0
## Median :2.600 Median : 88.00 Median :10.00 Median : 435.0
## Mean :2.668 Mean : 84.55 Mean :12.66 Mean : 483.5
## 3rd Qu.:3.220 3rd Qu.: 97.00 3rd Qu.:19.00 3rd Qu.: 608.0
## Max. :4.620 Max. :100.00 Max. :59.00 Max. :1570.0
## NA's :19
## mealcat collcat abv_hsg lgenroll
## Min. :1.000 Min. :1.00 Min. : 0.00 Min. :2.114
## 1st Qu.:1.000 1st Qu.:1.00 1st Qu.: 66.00 1st Qu.:2.505
## Median :2.000 Median :2.00 Median : 86.00 Median :2.638
## Mean :2.015 Mean :2.02 Mean : 78.75 Mean :2.640
## 3rd Qu.:3.000 3rd Qu.:3.00 3rd Qu.: 96.00 3rd Qu.:2.784
## Max. :3.000 Max. :3.00 Max. :100.00 Max. :3.196
## lm function
is used to fit models, carry out regression, analysis of variance and covariance.
Our first linear model
In our data example we are interested to study the relationship between student academic performance and characteristics of the school. For example we can use variable api00, a school-wide measure of academic performance, as the outcome, and variable enroll, the number of students in the school, as the predictor.
for our example, let us use the api00 for academic performance and full for emer, percentage of teachers with emergency credential.
making sure our data is an lm function
m1 <- lm(api00~ell
, data =d)
class( m1)## [1] "lm"print(m1)##
## Call:
## lm(formula = api00 ~ ell, data = d)
##
## Coefficients:
## (Intercept) ell
## 785.890 -4.396The estimated linear function is: \[ ap\hat{i}00 = 785.890 -4.396(ell) \] The coefficient for \(ell\) is -4.4, hence, for every one unit increase in ell, we would expect a 4.4 decrease in apio. Example, for a 200 english language learners, we would expect, on average, to have an api00 score 440 less than a 300 english language learners.
summary(m1)##
## Call:
## lm(formula = api00 ~ ell, data = d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -262.741 -63.605 1.443 68.242 212.310
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 785.890 7.370 106.64 <2e-16 ***
## ell -4.396 0.184 -23.89 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 91.28 on 398 degrees of freedom
## Multiple R-squared: 0.5893, Adjusted R-squared: 0.5882
## F-statistic: 571 on 1 and 398 DF, p-value: < 2.2e-16we can see in the summary that both the intercept and slope are statistically signficant.
Multiple R-squared of the model is equal to 0.5893 and adjust R-squared is 0.5992, which is adjusted for number of predictors.
In the simple linear regression model \(R^2\) is equal to the square of the correlation between the response and predictor variables. We can run the function cor() to confirm this.
r<-cor(d$ell,d$api00)r^2## [1] 0.5892615The last line gives the overall F-statistic, testing the fit of the current model against the null model, the model with only an intercept. F=44.83 with 1 and 398 degrees of freedom and with p-value equal to 7.339e-11. This p-value in a simple regression model is exactly equal to p-value of the slope, We can check the anova by using the function aov().
anova(m1)## Analysis of Variance Table
##
## Response: api00
## Df Sum Sq Mean Sq F value Pr(>F)
## ell 1 4757504 4757504 570.99 < 2.2e-16 ***
## Residuals 398 3316168 8332
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1As we said the output of the lm() function is an object of class lm. We can use functions ls() or str() to list the components of object m1.
ls(m1)## [1] "assign" "call" "coefficients" "df.residual"
## [5] "effects" "fitted.values" "model" "qr"
## [9] "rank" "residuals" "terms" "xlevels"m1$coefficients## (Intercept) ell
## 785.890265 -4.396082a vector of fitted values
m1$fitted.values[1:10]## 1 2 3 4 5 6 7 8
## 746.3255 693.5725 658.4039 667.1961 654.0078 772.7020 777.0981 772.7020
## 9 10
## 759.5138 719.9490Storing extracted components to new object
residuals <- m1$residgetting confident intervals for the coefficient of the model,
confint(m1)## 2.5 % 97.5 %
## (Intercept) 771.401848 800.378681
## ell -4.757761 -4.034403Using anova to extract sum of squares model
anova(m1)## Analysis of Variance Table
##
## Response: api00
## Df Sum Sq Mean Sq F value Pr(>F)
## ell 1 4757504 4757504 570.99 < 2.2e-16 ***
## Residuals 398 3316168 8332
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1The F-value in the anova states that the relationship has significance.
Another important function is predict() which can be use to get predicted value for new data points.
new.data <- data.frame(ell = c(500, 600, 700))
predict(m1, newdata = new.data)## 1 2 3
## -1412.151 -1851.759 -2291.367library(sjPlot)## Warning: package 'sjPlot' was built under R version 4.3.2## Learn more about sjPlot with 'browseVignettes("sjPlot")'.tab_model(m1)| api00 | |||
|---|---|---|---|
| Predictors | Estimates | CI | p |
| (Intercept) | 785.89 | 771.40 – 800.38 | <0.001 |
| ell | -4.40 | -4.76 – -4.03 | <0.001 |
| Observations | 400 | ||
| R2 / R2 adjusted | 0.589 / 0.588 | ||
Plotting a scatter plot and the regression line
plot(api00 ~ ell, data = d)
abline(m1, col = "blue") 
It appears in the figure that there are no outliers,
plot(api00 ~ ell, data = d)
text(d$ell, d$api00+20, labels = d$snum, cex = .7)
abline(m1, col = "blue")
library(ggplot2)## Warning: package 'ggplot2' was built under R version 4.3.2 ggplot(d, aes(x = ell, y = api00)) +
geom_point() +
stat_smooth(method = "lm", col = "red")## `geom_smooth()` using formula = 'y ~ x'
### Exercise 1 In the data set elemapi2v2 the variable full is the
percentage teachers with full credential for each school. Run the
regression model of api00 on full. Use this model to predict the mean of
api00 for full equal to 25%, 50%, 75%, and 90%.
Is the predicted value for full =25% valid?
Factor and dummy variable
In regression analysis, it is possible to include categorical predictors.
R uses object class factor to store and manipulate categorical variables. The lm function automatically treat variable type character as factor, but it is always safer to change the variable’s class to factor before the analysis. The function factor() is used to encode a variable (a vector) as a factor.
mealcat_F <-factor(d$mealcat)
str(mealcat_F)## Factor w/ 3 levels "1","2","3": 2 3 3 3 3 1 1 1 1 1 ...d$yr_rnd_F <- factor(d$yr_rnd)levels(d$yr_rnd_F)<-c("NO","Yes")table(d$yr_rnd_F)##
## NO Yes
## 308 92In R when we include a factor as a predictor to the model, the lm function by default generates dummy variables for each category of the factor. A dummy variable is a variable that takes values 0 and 1. If we are in that category the dummy variable is 1 and if we are not in that category the dummy variable is 0.The number of dummy variables is equal to the number of levels for the categorical variable minus 1. For example if we have 3 levels for the categorical variables, lm generates 2 dummy variables. If value for both dummy variable is 0 then we are in the third category. This third category is called the reference category.
###Regression of api00 on yr_rnd a two level categorical variable
m2<-lm(api00~yr_rnd_F, data=d)
summary(m2)##
## Call:
## lm(formula = api00 ~ yr_rnd_F, data = d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -273.539 -95.662 0.967 103.341 297.967
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 684.54 7.14 95.88 <2e-16 ***
## yr_rnd_FYes -160.51 14.89 -10.78 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 125.3 on 398 degrees of freedom
## Multiple R-squared: 0.226, Adjusted R-squared: 0.2241
## F-statistic: 116.2 on 1 and 398 DF, p-value: < 2.2e-16model.matrix(m2)[1:6,]## (Intercept) yr_rnd_FYes
## 1 1 0
## 2 1 0
## 3 1 0
## 4 1 0
## 5 1 0
## 6 1 0by writing out the regrresion equation, we have
\[api00 = 684.539 - 160.5064 \times yr\_rnd\_F(YES)\]
If a school is not a year-round school (i.e. yr_rnd_F is 0) the regression equation would simplify to
\[api00 = 684.539\]
If a school is a year-round school, the regression equation would simplify to \[api00=524.0326\] This indicates the mean of api00 given the school is not a year-round school is 524.0326.
In general when we have a two level categorical variable the intercept is the expected outcome for the reference group and the slope of the other category is the difference of expected outcome of that group with the reference category.
plot(api00~yr_rnd, data=d)
abline(m2)
Multiple Regression
Adding more predictors to a single regression model
Now, let’s look at an example of multiple regression, in which we have one outcome (dependent) variable and multiple predictors.
The percentage of variability of api00 explained by variable enroll was only 10.12%. In order to explain more variation, we can add more predictors. In R we can do this by adding variables with + to the formula of our lm() function. We add meals, the percentage of students who get full meals as an indicator of socioeconomic status, and full, the percentage of teachers with full credentials, to our model.
m3 <- lm(api00~enroll+meals+full, data=d)
summary(m3)##
## Call:
## lm(formula = api00 ~ enroll + meals + full, data = d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -181.721 -40.802 1.129 39.983 158.774
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 801.82983 26.42660 30.342 < 2e-16 ***
## enroll -0.05146 0.01384 -3.719 0.000229 ***
## meals -3.65973 0.10880 -33.639 < 2e-16 ***
## full 1.08109 0.23945 4.515 8.37e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 58.73 on 396 degrees of freedom
## Multiple R-squared: 0.8308, Adjusted R-squared: 0.8295
## F-statistic: 648.2 on 3 and 396 DF, p-value: < 2.2e-16anova(m3)## Analysis of Variance Table
##
## Response: api00
## Df Sum Sq Mean Sq F value Pr(>F)
## enroll 1 817326 817326 236.947 < 2.2e-16 ***
## meals 1 5820066 5820066 1687.263 < 2.2e-16 ***
## full 1 70313 70313 20.384 8.369e-06 ***
## Residuals 396 1365967 3449
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1We see that, on the F-test, the overall model is a significant improvement in fit compared to the intercept-only model. Also, all of the tests of the coefficients against the null value of zero are significant.
The R-square is 0.8308, meaning that approximately 83% of the variability of api00 is accounted for by the variables in the model. The adjusted R-square shows after taking the account of number of predictors in the model R_square is still about 0.83.
The coefficients for each of the variables indicates the amount of change one could expect in api00 given a one-unit increase in the value of that variable, given that all other variables in the model are held constant. For example, consider the variable meals. We would expect a decrease of about 3.66 in the api00 score for every one unit increase in percent free meals, assuming that all other variables in the model are held constant.
We see quite a difference in the coefficient of variable enroll compared to the simple linear regression. Before the coefficient for variable enroll was -.1999 and now it is -0.05146.
The ANOVA table shows the sum of squares explained by adding each variable sequentially to the model, or equivalently, the amount of sum of square residuals reduced by each additional variable.
For example variable enroll reduces the total error (RSS) by 817326. By adding variable meals we reduce additional 5820066 from the residual sum of squares and by adding variable full we reduce the error by 70313. Finally we have 1365967 left as unexplained error. The total sum of squares (TSS) is the sum of all of the sums of squares added together. To get the total sum of square of variable api00 we can multiply its’ variance by (n−1).
sum(anova(m3)$Sum)## [1] 8073672(400-1)*var(d$api00)## [1] 8073672Standardized regression coefficients
Some researchers are interested in comparing the relative strength of the various predictors within the model.To address this problem we use standardized regression coefficients, which can be obtain by transforming the outcome and predictor variables all to their standardized scores, also called z-scores, before running the regression.
m3.sd <- lm(scale(api00) ~ scale(enroll) + scale(meals) + scale(full), data = d)
summary(m3.sd)##
## Call:
## lm(formula = scale(api00) ~ scale(enroll) + scale(meals) + scale(full),
## data = d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.27749 -0.28683 0.00793 0.28108 1.11617
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.454e-16 2.064e-02 0.000 1.000000
## scale(enroll) -8.191e-02 2.203e-02 -3.719 0.000229 ***
## scale(meals) -8.210e-01 2.441e-02 -33.639 < 2e-16 ***
## scale(full) 1.136e-01 2.517e-02 4.515 8.37e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4129 on 396 degrees of freedom
## Multiple R-squared: 0.8308, Adjusted R-squared: 0.8295
## F-statistic: 648.2 on 3 and 396 DF, p-value: < 2.2e-16In the standardized regression coefficients summary we see that the intercept is zero and all t-statistics (p-values) for the other coefficients are exactly the same as the original model.
Because the coefficients are all in the same standardized units, standard deviations, you can compare these coefficients to assess the relative strength of each of the predictors. In this example, meals has the largest Beta coefficient, -0.821.
Thus, a one standard deviation increase in meals leads to a 0.821 standard deviation decrease in predicted api00, with the other variables held constant.
Regression model with interaction between two continuous predictors
In this topic, we consider the interaction between the predictors
m4 <- lm(api00 ~ enroll + meals + enroll:meals , data = d)
summary(m4) ##
## Call:
## lm(formula = api00 ~ enroll + meals + enroll:meals, data = d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -211.186 -38.834 -1.137 38.997 163.713
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.833e+02 1.665e+01 53.034 <2e-16 ***
## enroll 8.835e-04 3.362e-02 0.026 0.9790
## meals -3.425e+00 2.344e-01 -14.614 <2e-16 ***
## enroll:meals -9.537e-04 4.292e-04 -2.222 0.0269 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 59.85 on 396 degrees of freedom
## Multiple R-squared: 0.8243, Adjusted R-squared: 0.823
## F-statistic: 619.3 on 3 and 396 DF, p-value: < 2.2e-16The intercept interpreted as before and is the expected api00 when both enroll and meals are set to zero.
The coefficient of enroll is interpreted as increase of the mean of api00 by 0.0008835 for one unit increase of number of enrollment given the value of meals is zero.
The coefficient of meals is interpreted as average decrease of the mean of api00 by 3.425 for one unit increase of meals (one percentage increase) given the value of enroll is zero.
Adding interaction term means that the effect of enroll and meals is no longer constant for different values of the other predictor.
The interaction term can be interpreted in two ways.
If we use enroll as moderator variable then we can say that the effect of meals changes by −0.0009537 for one unit increase of enroll.
If we use meals as moderator variable then we can say that the effect of enroll changes by −0.0009537 for one unit increase of meals.
Regression model with interaction between two categorical variables
In here, instead of two continuous variables, we consider the interaction between the two categorical variables
d$mealcat_F <- relevel(mealcat_F, ref="3")m6 <- lm(api00 ~ d$yr_rnd_F*mealcat_F, data = d)
summary(m6) ##
## Call:
## lm(formula = api00 ~ d$yr_rnd_F * mealcat_F, data = d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -207.533 -50.764 -1.843 48.874 179.000
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 521.493 8.414 61.978 < 2e-16 ***
## d$yr_rnd_FYes -33.493 11.771 -2.845 0.00467 **
## mealcat_F1 288.193 10.443 27.597 < 2e-16 ***
## mealcat_F2 123.781 10.552 11.731 < 2e-16 ***
## d$yr_rnd_FYes:mealcat_F1 -40.764 29.231 -1.395 0.16394
## d$yr_rnd_FYes:mealcat_F2 -18.248 22.256 -0.820 0.41278
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 68.87 on 394 degrees of freedom
## Multiple R-squared: 0.7685, Adjusted R-squared: 0.7656
## F-statistic: 261.6 on 5 and 394 DF, p-value: < 2.2e-16The intercept interpreted as the expected api00 when both yr_rnd_F and mealcat_F are at their reference level. Here, it is the expected api00 given school is not year round and the meals category is at level “3”.
The coefficient of yr_rnd_FYes is the difference of the expected api00 for year round school with school is not year round when meals category is at level “3”
The coefficient of mealcat_F1 is the difference of the expected api00 between mealcat_F at level “1” and mealcat_F at level “3” given school is not year round.
The coefficient of mealcat_F2 is the difference of the expected api00 between mealcat_F at level “2” and mealcat_F at level “3” given schoo is not year round.
The interaction terms can be interpreted in two ways. Here we only interpret it in one way:
The coefficient of yr_rnd_FYes:mealcat_F1 is the expected difference api00 for year round school with school is not year round if the meals category is “1” minus the expected difference api00 for year round school with school is not year round if the meals category is “3”.
The coefficient of yr_rnd_FYes:mealcat_F2 is the expected api00 for year round school with school is not year round if the meals category is “2” minus the expected api00 for year round school with school is not year round if the meals category is “3”.
For example the expected mean of api00 for a school that is year round with the mealcat equal to “2” is:
\[521.493+(-33.493)+123.781+(-18.248) = 593.533\tag{5.2.1}\] The expected mean of api00 for a school that is not year round with the mealcat equals to “2” is:
\[521+123.781=645.274 \tag{5.2.2}\] Thus, the expected difference api00 for year round school with school is not year round if the meals category equals to “2” is:
in (5.2.1) the expected mean of apio00 for a school that is year
round with mealcat = 2, and in (5.2.2) is the expected mean of api00 for
a school that is not a year round with the mealcat = 2, Here we subtract
the sum in (5.2.1) and (5.2.2)
\[593.533-645.274 = -51.731
\tag{5.2.3}\] The difference of the expected api00 for year round
school with school is not year round when meals category is at level “3
is the coefficient of yr_rnd_FYes = -33.493
If we calculate the difference between the above differences we get: \[-51.741-(-33.493) = -18.248 \tag{5.2.4}\] which is the coefficient for yr_rnd_FYes:mealcat_F2.
If we need to test overall effect of interaction terms (i.e. simultaneously test both interaction coefficients equal to zero) we can use likelihood ratio F test using anova function. To do so, we run a model without interaction terms and use anova function to test the difference between two models.
m0 <- lm(api00 ~ yr_rnd_F + mealcat_F , data = d)
anova(m0, m6)## Analysis of Variance Table
##
## Model 1: api00 ~ yr_rnd_F + mealcat_F
## Model 2: api00 ~ d$yr_rnd_F * mealcat_F
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 396 1879528
## 2 394 1868944 2 10584 1.1156 0.3288The F test means that we do not have enough evidence to reject the base model m0 and therefore, the effect yr_rnd unlikely change for different levels of mealcat.
Regression model with interaction between continuous predictor and a categorical predictor
Here we consider two predictors, one is continuous and the other is categorical, in the above example, let us use enroll for continuous and yr_rnd_F for categorical, for their interactions, we simply multiply the two predictors in the linear model formula
m7 <- lm(api00 ~ yr_rnd_F * enroll , data = d)
summary(m7) ##
## Call:
## lm(formula = api00 ~ yr_rnd_F * enroll, data = d)
##
## Residuals:
## Min 1Q Median 3Q Max
## -274.043 -94.781 0.417 97.666 309.573
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 682.068556 18.797436 36.285 <2e-16 ***
## yr_rnd_FYes -74.858236 48.224281 -1.552 0.1214
## enroll 0.006021 0.042396 0.142 0.8871
## yr_rnd_FYes:enroll -0.120218 0.072075 -1.668 0.0961 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 125 on 396 degrees of freedom
## Multiple R-squared: 0.2335, Adjusted R-squared: 0.2277
## F-statistic: 40.21 on 3 and 396 DF, p-value: < 2.2e-16The intercept interpreted as before and is the expected api00 when enroll is zero and the school is not year round.
The coefficient of enroll is interpreted as increase of the mean of api00 by for one unit increase of number of enrollment given the school is not year round.
The coefficient of yr_rnd_FYes is interpreted as the difference between the expected api00 for year round school and not year round school when given the value of enroll is zero.
The interaction term can be interpreted as: The change of the effect of enroll to the expected mean api00 when the school is year round to when the school is not year round.
Plotting the interaction between a continuous predictor and categorical predictor
Sometimes it is a good idea to plot the predicted values for different levels of predictors to visualize the interaction. We use function predict for a new data to plot this interaction.
First get the range of continuous predictor
range(d$enroll)## [1] 130 1570#make a sequence of number of enroll from range of enroll 130-1570 increment by 5
new.enroll <- seq(130, 1570, 5)
new.yr_rnd <- rep(levels(d$yr_rnd_F), each = length(new.enroll))
new.data.plot <- data.frame(enroll = rep(new.enroll, times = 2), yr_rnd_F = new.yr_rnd)
new.data.plot$predicted_api00 <- predict(m7, newdata = new.data.plot)
library(ggplot2)
ggplot(new.data.plot, aes(x = enroll, y = predicted_api00, colour = yr_rnd_F)) +
geom_line(lwd=1.2)
Regression Diagnostics
Introduction
In the previous part, we learned how to do ordinary linear regression with R. Without verifying that the data have met the assumptions underlying OLS regression, results of regression analysis may be misleading. Here will explore how you can use R to check on how well your data meet the assumptions of OLS regression. In particular, we will consider the following assumptions.
Homogeneity of variance (homoscedasticity): The error variance should be constant
Linearity: the relationships between the predictors and the outcome variable should be linear.
Independence: The errors associated with one observation are not correlated with the errors of any other observation
Normality: the errors should be normally distributed. Technically normality is necessary only for hypothesis tests to be valid.
Model specification: The model should be properly specified (including all relevant variables, and excluding irrelevant variables)
Additionally, there are issues that can arise during the analysis that, while strictly speaking are not assumptions of regression, are none the less, of great concern to data analysts.
Influence: individual observations that exert undue influence on the coefficients
Collinearity: predictors that are highly collinear, i.e., linearly related, can cause problems in estimating the regression coefficients.
Many graphical methods and numerical tests have been developed over the years for regression diagnostics.
R has many of these methods in stats package which is already installed and loaded in R. There are some other tools in different packages that we can use by installing and loading those packages in our R environment.
###unusual and Influential data
A single observation that is substantially different from all other observations can make a large difference in the results of your regression analysis. If a single observation (or small group of observations) substantially changes your results, you would want to know about this and investigate further. There are three ways that an observation can be unusual.
Outliers: In linear regression, an outlier is an observation with large residual. In other words, it is an observation whose dependent-variable value is unusual given its values on the predictor variables. An outlier may indicate a sample peculiarity or may indicate a data entry error or other problem.
Leverage: An observation with an extreme value on a predictor variable is called a point with high leverage. Leverage is a measure of how far an observation deviates from the mean of that variable. These leverage points can have an effect on the estimate of regression coefficients.
Influence: An observation is said to be influential if removing the observation substantially changes the estimate of coefficients. Influence can be thought of as the product of leverage and outlierness.
library(car)## Warning: package 'car' was built under R version 4.3.2## Loading required package: carData## Warning: package 'carData' was built under R version 4.3.2m9<-lm(api00~enroll+meals+full, data=d)
scatterplotMatrix(~api00 + enroll + meals+full, data=d)
The graphs of api00 with other variables show some potential problems. In every plot, we seeone or more data point that is far away from the rest of the data points.
Studentized residuals can be used to identify outliers. In R we use rstandard() function tocompute Studentized residuals.
res.std <- rstandard(m9)
plot(res.std, ylab="Standardized Residual", ylim=c(-3.5,3.5))
abline(h =c(-3,0,3),lty =2)
index <- which(res.std > 3 | res.std < -3)
text(index-20, res.std[index] , labels = d$snum[index])
print(index)## 226
## 226print(d$snum[index])## [1] 211We should pay attention to studentized residuals that exceed +2 or -2, and get even more concerned about residuals that exceed +2.5 or -2.5 and even yet more concerned about residuals that exceed +3 or -3. These results show that school number 211 is the most worrisome observation.
outlierTest(m9)## No Studentized residuals with Bonferroni p < 0.05
## Largest |rstudent|:
## rstudent unadjusted p-value Bonferroni p
## 226 -3.151186 0.00175 0.70001library(faraway)## Warning: package 'faraway' was built under R version 4.3.2##
## Attaching package: 'faraway'## The following objects are masked from 'package:car':
##
## logit, vifh <- influence(m9)$hat
halfnorm(influence(m9)$hat, ylab = "leverage")
Cook’s distance is a measure for influence points. A point with high level of cook’s distance is considers as a point with high influence point. A cut of value for cook’s distance can be calculated as \[\frac{4}{n-p-1}\] Where n is sample size and p is number of predictors. We can plot Cook’s distance using the following code
cutoff <- 4/((nrow(d)-length(m2$coefficients)-2))
plot(m9, which = 4, cook.levels = cutoff)
We can use influencePlot() function in package “car” to identify influence point. It plots Studentized residuals against leverage with cook’s distance.
influencePlot(m9, main="Influence Plot",
sub="Circle size is proportional to Cook's Distance" )
## StudRes Hat CookD
## 8 0.18718812 0.08016299 7.652779e-04
## 93 2.76307269 0.02940688 5.687488e-02
## 210 0.03127861 0.06083329 1.588292e-05
## 226 -3.15118603 0.01417076 3.489753e-02
## 346 -2.83932062 0.00412967 8.211170e-03infIndexPlot(m9)
### Checking Homoscedasticity
One of the main assumptions for the ordinary least squares regression is the homogeneity of variance of the residuals. If the model is well-fitted, there should be no pattern to the residuals plotted against the fitted values. If the variance of the residuals is non-constant then the residual variance is said to be “heteroscedastic.” There are graphical and non-graphical methods for detecting heteroscedasticity. A commonly used graphical method is to plot the residuals versus fitted (predicted) values.
plot(m9$resid ~ m9$fitted.values)
abline(h = 0, lty = 2)
### Checking Linearity To check linearity residuals should be plotted
against the fit as well as other predictors. If any of these plots show
systematic shapes, then the linear model is not appropriate and some
nonlinear terms may need to be added.
residualPlots(m9)
## Test stat Pr(>|Test stat|)
## enroll 0.0111 0.9911
## meals -0.6238 0.5331
## full 1.1565 0.2482
## Tukey test -0.8411 0.4003Issues on independence
A simple visual check would be to plot the residuals versus the time variable.
plot(m9$resid ~ d$snum)
### Checking Normality Residuals
Normality of residuals is only required for valid hypothesis testing, that is, the normality assumption assures that the p-values for the t-tests and F-test will be valid. Normality is not required in order to obtain unbiased estimates of the regression coefficients.
OLS regression merely requires that the residuals (errors) be identically and independently distributed. Furthermore, there is no assumption or requirement that the predictor variables be normally distributed. If this were the case than we would not be able to use dummy coded variables in our models.
because of large sample theory if we have large enough sample size we do not even need the residuals be normally distributed, however, for small sample sizes, normality is required.
qqnorm(m9$resid)
qqline(m9$resid)
Checking for Multicollinearity
When there is a perfect linear relationship among the predictors, the estimates for a regression model cannot be uniquely computed. The term collinearity implies that two variables are near perfect linear combinations of one another. When more than two variables are involved it is often called multicollinearity, although the two terms are often used interchangeably.
The primary concern is that as the degree of multicollinearity increases, the regression model estimates of the coefficients become unstable and the standard errors for the coefficients can get wildly inflated.
VIF, variance inflation factor, is used to measure the degree of multicollinearity. As a rule of thumb, a variable whose VIF values are greater than 10 may merit further investigation. Tolerance, defined as 1/VIF, is used by many researchers to check on the degree of collinearity. A tolerance value lower than 0.1 is comparable to a VIF of 10. It means that the variable could be considered as a linear combination of other independent variables.
car::vif(m9)## enroll meals full
## 1.135733 1.394279 1.482305Model Specification
A model specification error can occur when one or more relevant variables are omitted from the model or one or more irrelevant variables are included in the model.
If relevant variables are omitted from the model, the common variance they share with included variables may be wrongly attributed to those variables, and the error term is inflated. On the other hand, if irrelevant variables are included in the model, the common variance they share with included variables may be wrongly attributed to them. Model specification errors can substantially affect the estimate of regression coefficients.
avPlots(m9)