Untitled
Nguyen_LSCM
1/8/2022
Correlation and regression fundamentals with tidy data principles
LEARNING OBJECTIVE
Analyze the results of correlationship tests and simple regression models for many data sets at once
INTRODUCTION
This article only requires the tidymodels package.
While the tidymodels package broom is useful for summarizing the result of a single analysis in a consistent format, it is really designed for high-throughput applications, where you must combine results from multiple analyses. These could be subgroups of data, analyses using different models, bootstrap replicates, permutations, and so on. In particular, it plays well with ther nest() / unnest() function from tidyr and the map() function in purr
CORRELATION ANALYSIS
suppressMessages(library(tidymodels))
data("Orange")
Orange <- as_tibble(Orange)
Orange## # A tibble: 35 x 3
## Tree age circumference
## <ord> <dbl> <dbl>
## 1 1 118 30
## 2 1 484 58
## 3 1 664 87
## 4 1 1004 115
## 5 1 1231 120
## 6 1 1372 142
## 7 1 1582 145
## 8 2 118 33
## 9 2 484 69
## 10 2 664 111
## # ... with 25 more rowsThis contains 35 observations of three variables: Tree, age, and circumference. Tree is a factor with 5 levels describing five trees. As might be expected, age and circumference are correlated:
theme_set(theme_light())
cor(Orange$age, Orange$circumference)## [1] 0.9135189suppressMessages(library(ggplot2))
ggplot(Orange, aes(age, circumference, color = Tree)) +
geom_line()
Suppose you want to test for correlations individually within each tree. You can do this with dplyr’s group_by:
Orange %>%
group_by(Tree) %>%
summarize(correlation = cor(age, circumference))## # A tibble: 5 x 2
## Tree correlation
## <ord> <dbl>
## 1 3 0.988
## 2 1 0.985
## 3 5 0.988
## 4 2 0.987
## 5 4 0.984(Note that the correlations are much higher than the aggregated one, and also we can now see the correlation is simular across trees).
Suppose that instead of simply estimating a correlation, we want to perform a hypothesis test with cor.test():
ct <- cor.test(Orange$age, Orange$circumference)
ct##
## Pearson's product-moment correlation
##
## data: Orange$age and Orange$circumference
## t = 12.9, df = 33, p-value = 1.931e-14
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.8342364 0.9557955
## sample estimates:
## cor
## 0.9135189This test output comtains multiple values we may be interested in. Some are vectors of length 1, such as p-value and the estimate, and some are longer, such as the confidence interval. We can get this into a nicely organized tibble using the tidy() function:
tidy(ct)## # A tibble: 1 x 8
## estimate statistic p.value parameter conf.low conf.high method alternative
## <dbl> <dbl> <dbl> <int> <dbl> <dbl> <chr> <chr>
## 1 0.914 12.9 1.93e-14 33 0.834 0.956 Pearson'~ two.sidedOften, we want to perform multiple tests or fit multiple models, each on a different part of the data. In this case, we recommended a nest-map-unnest workflow. For example, suppose we want to perform correlation tests for each different tree. We start by nesting our data based on the group of interest:
nested <-
Orange %>%
nest(data = c(age, circumference))
nested %>% unnest(data)## # A tibble: 35 x 3
## Tree age circumference
## <ord> <dbl> <dbl>
## 1 1 118 30
## 2 1 484 58
## 3 1 664 87
## 4 1 1004 115
## 5 1 1231 120
## 6 1 1372 142
## 7 1 1582 145
## 8 2 118 33
## 9 2 484 69
## 10 2 664 111
## # ... with 25 more rowsThen we perform a correlation test for each nested tibble using purrr:map():
nested %>%
mutate(test = map(data, ~ cor.test(.x$age, .x$circumference)))## # A tibble: 5 x 3
## Tree data test
## <ord> <list> <list>
## 1 1 <tibble [7 x 2]> <htest>
## 2 2 <tibble [7 x 2]> <htest>
## 3 3 <tibble [7 x 2]> <htest>
## 4 4 <tibble [7 x 2]> <htest>
## 5 5 <tibble [7 x 2]> <htest>This results in a list-column of S3 objects. We want to tidy each of the objects, which we can also do with map()
nested %>%
mutate(
test = map(data, ~ cor.test(.x$age, .x$circumference)),
tidided = map(test, tidy)
)## # A tibble: 5 x 4
## Tree data test tidided
## <ord> <list> <list> <list>
## 1 1 <tibble [7 x 2]> <htest> <tibble [1 x 8]>
## 2 2 <tibble [7 x 2]> <htest> <tibble [1 x 8]>
## 3 3 <tibble [7 x 2]> <htest> <tibble [1 x 8]>
## 4 4 <tibble [7 x 2]> <htest> <tibble [1 x 8]>
## 5 5 <tibble [7 x 2]> <htest> <tibble [1 x 8]>Finally, we want to unnest the tidied data frames so we can see the results in a flat tibble. All together, this looks like:
Orange %>%
nest(data = c(age, circumference)) %>%
mutate(
test = map(data, ~ cor.test(.x$age, .x$circumference)),
tidied = map(test, tidy)
) %>%
unnest(cols = tidied) %>%
select(-data, -test)## # A tibble: 5 x 9
## Tree estimate statistic p.value parameter conf.low conf.high method
## <ord> <dbl> <dbl> <dbl> <int> <dbl> <dbl> <chr>
## 1 1 0.985 13.0 0.0000485 5 0.901 0.998 Pearson's pro~
## 2 2 0.987 13.9 0.0000343 5 0.914 0.998 Pearson's pro~
## 3 3 0.988 14.4 0.0000290 5 0.919 0.998 Pearson's pro~
## 4 4 0.984 12.5 0.0000573 5 0.895 0.998 Pearson's pro~
## 5 5 0.988 14.1 0.0000318 5 0.916 0.998 Pearson's pro~
## # ... with 1 more variable: alternative <chr>REGRESSION MODELS
This type of workflow becomes even more useful when appleid to regressions. Untidy output for a regression looks like:
lm_fit <- lm(age ~ circumference, data = Orange)
summary(lm_fit)##
## Call:
## lm(formula = age ~ circumference, data = Orange)
##
## Residuals:
## Min 1Q Median 3Q Max
## -317.88 -140.90 -17.20 96.54 471.16
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.6036 78.1406 0.212 0.833
## circumference 7.8160 0.6059 12.900 1.93e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 203.1 on 33 degrees of freedom
## Multiple R-squared: 0.8345, Adjusted R-squared: 0.8295
## F-statistic: 166.4 on 1 and 33 DF, p-value: 1.931e-14When we tidy the results, we get multiple rows of output for each model:
tidy(lm_fit)## # A tibble: 2 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 16.6 78.1 0.212 8.33e- 1
## 2 circumference 7.82 0.606 12.9 1.93e-14Now we can handle multiple regressions at once using exactly the same workflow as before:
Orange %>%
nest(data = c(-Tree)) %>%
mutate(
fit = map(data, ~ lm(age ~ circumference, data = .x)),
tidied = map(fit, tidy)
) %>%
unnest(tidied) %>%
select(-data, -fit)## # A tibble: 10 x 6
## Tree term estimate std.error statistic p.value
## <ord> <chr> <dbl> <dbl> <dbl> <dbl>
## 1 1 (Intercept) -265. 98.6 -2.68 0.0436
## 2 1 circumference 11.9 0.919 13.0 0.0000485
## 3 2 (Intercept) -132. 83.1 -1.59 0.172
## 4 2 circumference 7.80 0.560 13.9 0.0000343
## 5 3 (Intercept) -210. 85.3 -2.46 0.0574
## 6 3 circumference 12.0 0.835 14.4 0.0000290
## 7 4 (Intercept) -76.5 88.3 -0.867 0.426
## 8 4 circumference 7.17 0.572 12.5 0.0000573
## 9 5 (Intercept) -54.5 76.9 -0.709 0.510
## 10 5 circumference 8.79 0.621 14.1 0.0000318You can just as easily use multiple predictors in the regressions, as shown here on the mtcars dataset. We nest the data into automatic vs manual cars (the am column), then perform the regression within each nested tibble.
data(mtcars)
mtcars %>% as_tibble(mtcars)## # A tibble: 32 x 11
## mpg cyl disp hp drat wt qsec vs am gear carb
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 21 6 160 110 3.9 2.62 16.5 0 1 4 4
## 2 21 6 160 110 3.9 2.88 17.0 0 1 4 4
## 3 22.8 4 108 93 3.85 2.32 18.6 1 1 4 1
## 4 21.4 6 258 110 3.08 3.22 19.4 1 0 3 1
## 5 18.7 8 360 175 3.15 3.44 17.0 0 0 3 2
## 6 18.1 6 225 105 2.76 3.46 20.2 1 0 3 1
## 7 14.3 8 360 245 3.21 3.57 15.8 0 0 3 4
## 8 24.4 4 147. 62 3.69 3.19 20 1 0 4 2
## 9 22.8 4 141. 95 3.92 3.15 22.9 1 0 4 2
## 10 19.2 6 168. 123 3.92 3.44 18.3 1 0 4 4
## # ... with 22 more rowsmtcars## mpg cyl disp hp drat wt qsec vs am gear carb
## Mazda RX4 21.0 6 160.0 110 3.90 2.620 16.46 0 1 4 4
## Mazda RX4 Wag 21.0 6 160.0 110 3.90 2.875 17.02 0 1 4 4
## Datsun 710 22.8 4 108.0 93 3.85 2.320 18.61 1 1 4 1
## Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215 19.44 1 0 3 1
## Hornet Sportabout 18.7 8 360.0 175 3.15 3.440 17.02 0 0 3 2
## Valiant 18.1 6 225.0 105 2.76 3.460 20.22 1 0 3 1
## Duster 360 14.3 8 360.0 245 3.21 3.570 15.84 0 0 3 4
## Merc 240D 24.4 4 146.7 62 3.69 3.190 20.00 1 0 4 2
## Merc 230 22.8 4 140.8 95 3.92 3.150 22.90 1 0 4 2
## Merc 280 19.2 6 167.6 123 3.92 3.440 18.30 1 0 4 4
## Merc 280C 17.8 6 167.6 123 3.92 3.440 18.90 1 0 4 4
## Merc 450SE 16.4 8 275.8 180 3.07 4.070 17.40 0 0 3 3
## Merc 450SL 17.3 8 275.8 180 3.07 3.730 17.60 0 0 3 3
## Merc 450SLC 15.2 8 275.8 180 3.07 3.780 18.00 0 0 3 3
## Cadillac Fleetwood 10.4 8 472.0 205 2.93 5.250 17.98 0 0 3 4
## Lincoln Continental 10.4 8 460.0 215 3.00 5.424 17.82 0 0 3 4
## Chrysler Imperial 14.7 8 440.0 230 3.23 5.345 17.42 0 0 3 4
## Fiat 128 32.4 4 78.7 66 4.08 2.200 19.47 1 1 4 1
## Honda Civic 30.4 4 75.7 52 4.93 1.615 18.52 1 1 4 2
## Toyota Corolla 33.9 4 71.1 65 4.22 1.835 19.90 1 1 4 1
## Toyota Corona 21.5 4 120.1 97 3.70 2.465 20.01 1 0 3 1
## Dodge Challenger 15.5 8 318.0 150 2.76 3.520 16.87 0 0 3 2
## AMC Javelin 15.2 8 304.0 150 3.15 3.435 17.30 0 0 3 2
## Camaro Z28 13.3 8 350.0 245 3.73 3.840 15.41 0 0 3 4
## Pontiac Firebird 19.2 8 400.0 175 3.08 3.845 17.05 0 0 3 2
## Fiat X1-9 27.3 4 79.0 66 4.08 1.935 18.90 1 1 4 1
## Porsche 914-2 26.0 4 120.3 91 4.43 2.140 16.70 0 1 5 2
## Lotus Europa 30.4 4 95.1 113 3.77 1.513 16.90 1 1 5 2
## Ford Pantera L 15.8 8 351.0 264 4.22 3.170 14.50 0 1 5 4
## Ferrari Dino 19.7 6 145.0 175 3.62 2.770 15.50 0 1 5 6
## Maserati Bora 15.0 8 301.0 335 3.54 3.570 14.60 0 1 5 8
## Volvo 142E 21.4 4 121.0 109 4.11 2.780 18.60 1 1 4 2mtcars %>%
nest(data = -c(am)) %>%
mutate(
fit = map(data, ~ lm (wt ~ mpg + qsec + gear, data = .x)),
tidied = map(fit, tidy)
) %>%
unnest(tidied) %>%
select(-data, -fit)## # A tibble: 8 x 6
## am term estimate std.error statistic p.value
## <dbl> <chr> <dbl> <dbl> <dbl> <dbl>
## 1 1 (Intercept) 4.28 3.46 1.24 0.247
## 2 1 mpg -0.101 0.0294 -3.43 0.00750
## 3 1 qsec 0.0398 0.151 0.264 0.798
## 4 1 gear -0.0229 0.349 -0.0656 0.949
## 5 0 (Intercept) 4.92 1.40 3.52 0.00309
## 6 0 mpg -0.192 0.0443 -4.33 0.000591
## 7 0 qsec 0.0919 0.0983 0.935 0.365
## 8 0 gear 0.147 0.368 0.398 0.696What if you want not just the tidy() output, but the augment() and glance() outputs as well, while still performing each regression only once? Since we’re using list-columns, we can just fit the model once and use multiple list-columns to store the tidied, glanced and augmented outputs.
regressions <-
mtcars %>%
nest(data = c(-am)) %>%
mutate(
fit = map(data, ~lm(wt ~ mpg + qsec + gear, data = .x)),
tidied = map(fit, tidy),
glanced = map(fit, glance),
augmented = map(fit, augment)
)
regressions## # A tibble: 2 x 6
## am data fit tidied glanced augmented
## <dbl> <list> <list> <list> <list> <list>
## 1 1 <tibble [13 x 10]> <lm> <tibble [4 x 5]> <tibble [1 x 12]> <tibble [1~
## 2 0 <tibble [19 x 10]> <lm> <tibble [4 x 5]> <tibble [1 x 12]> <tibble [1~regressions %>%
select(tidied) %>%
unnest(tidied)## # A tibble: 8 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 4.28 3.46 1.24 0.247
## 2 mpg -0.101 0.0294 -3.43 0.00750
## 3 qsec 0.0398 0.151 0.264 0.798
## 4 gear -0.0229 0.349 -0.0656 0.949
## 5 (Intercept) 4.92 1.40 3.52 0.00309
## 6 mpg -0.192 0.0443 -4.33 0.000591
## 7 qsec 0.0919 0.0983 0.935 0.365
## 8 gear 0.147 0.368 0.398 0.696regressions %>%
select(glanced) %>%
unnest(glanced)## # A tibble: 2 x 12
## r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.833 0.778 0.291 15.0 0.000759 3 -0.00580 10.0 12.8
## 2 0.625 0.550 0.522 8.32 0.00170 3 -12.4 34.7 39.4
## # ... with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>regressions %>%
select(augmented) %>%
unnest(augmented)## # A tibble: 32 x 10
## wt mpg qsec gear .fitted .resid .hat .sigma .cooksd .std.resid
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2.62 21 16.5 4 2.73 -0.107 0.517 0.304 0.0744 -0.527
## 2 2.88 21 17.0 4 2.75 0.126 0.273 0.304 0.0243 0.509
## 3 2.32 22.8 18.6 4 2.63 -0.310 0.312 0.279 0.188 -1.29
## 4 2.2 32.4 19.5 4 1.70 0.505 0.223 0.233 0.278 1.97
## 5 1.62 30.4 18.5 4 1.86 -0.244 0.269 0.292 0.0889 -0.982
## 6 1.84 33.9 19.9 4 1.56 0.274 0.286 0.286 0.125 1.12
## 7 1.94 27.3 18.9 4 2.19 -0.253 0.151 0.293 0.0394 -0.942
## 8 2.14 26 16.7 5 2.21 -0.0683 0.277 0.307 0.00732 -0.276
## 9 1.51 30.4 16.9 5 1.77 -0.259 0.430 0.284 0.263 -1.18
## 10 3.17 15.8 14.5 5 3.15 0.0193 0.292 0.308 0.000644 0.0789
## # ... with 22 more rowssessioninfo::session_info()## - Session info ---------------------------------------------------------------
## setting value
## version R version 4.1.1 (2021-08-10)
## os Windows 10 x64
## system x86_64, mingw32
## ui RTerm
## language (EN)
## collate English_United States.1252
## ctype English_United States.1252
## tz Asia/Bangkok
## date 2022-01-11
##
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## package * version date lib
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## CRAN (R 4.1.1)
## CRAN (R 4.1.1)
## CRAN (R 4.1.1)
## CRAN (R 4.1.0)
## CRAN (R 4.1.1)
##
## [1] D:/R/R-4.1.1/library