# Linear Regression model

## Highlights & Limitations

• Supports prediction intervals, it uses the qr.solve() function to parse the interval coefficient of each term.
• Supports categorical variables and interactions
• Only treatment contrast (contr.treatment) are supported.
• offset is supported
• Categorical variables are supported
• In-line functions in the formulas are not supported:
• OK - wt ~ mpg + am
• OK - mutate(mtcars, newam = paste0(am)) and then wt ~ mpg + newam
• Not OK - wt ~ mpg + as.factor(am)
• Not OK - wt ~ mpg + as.character(am)

## How it works

df <- mtcars %>%
mutate(char_cyl = paste0("cyl", cyl)) %>%
select(mpg, wt, char_cyl, am)

model <- lm(mpg ~ wt + char_cyl, offset = am, data = df)

It returns a SQL query that contains the coefficients (model\$coefficients) operated against the correct variable or categorical variable value. In most cases the resulting SQL is one short CASE WHEN statement per coefficient. It appends the offset field or value, if one is provided.

library(tidypredict)

tidypredict_sql(model, dbplyr::simulate_mssql())
## <SQL> ((((32.4105336886021) + ((wt) * (-2.83243330448326))) + ((CASE WHEN (((char_cyl) = ('cyl6')) =  'TRUE') THEN (1.0) WHEN (((char_cyl) = ('cyl6')) =  'FALSE') THEN (0.0) END) * (-4.26714873091281))) + ((CASE WHEN (((char_cyl) = ('cyl8')) =  'TRUE') THEN (1.0) WHEN (((char_cyl) = ('cyl8')) =  'FALSE') THEN (0.0) END) * (-6.12588309683682))) + (am)

Alternatively, use tidypredict_to_column() if the results are the be used or previewed in dplyr.

df %>%
tidypredict_to_column(model) %>%
head(10) 
##     mpg    wt char_cyl am      fit
## 1  21.0 2.620     cyl6  1 21.72241
## 2  21.0 2.875     cyl6  1 21.00014
## 3  22.8 2.320     cyl4  1 26.83929
## 4  21.4 3.215     cyl6  0 19.03711
## 5  18.7 3.440     cyl8  0 16.54108
## 6  18.1 3.460     cyl6  0 18.34317
## 7  14.3 3.570     cyl8  0 16.17286
## 8  24.4 3.190     cyl4  0 23.37507
## 9  22.8 3.150     cyl4  0 23.48837
## 10 19.2 3.440     cyl6  0 18.39981

## Prediction intervals

Use tidypredict_sql_interval() to get the SQL query that operates the prediction interval. The interval defaults to 0.95

tidypredict_sql_interval(model, dbplyr::simulate_mssql())
## <SQL> (2.04840714179524) * SQRT((((((-0.176776695296637) * (-0.176776695296637) * (6.63799055122669)) + (((-0.590557271637747) + ((wt) * (0.183559646169165))) * ((-0.590557271637747) + ((wt) * (0.183559646169165))) * (6.63799055122669))) + ((((-0.126215672528828) + ((wt) * (0.0101118696567173))) + ((CASE WHEN (((char_cyl) = ('cyl6')) =  'TRUE') THEN (1.0) WHEN (((char_cyl) = ('cyl6')) =  'FALSE') THEN (0.0) END) * (0.428266330860589))) * (((-0.126215672528828) + ((wt) * (0.0101118696567173))) + ((CASE WHEN (((char_cyl) = ('cyl6')) =  'TRUE') THEN (1.0) WHEN (((char_cyl) = ('cyl6')) =  'FALSE') THEN (0.0) END) * (0.428266330860589))) * (6.63799055122669))) + (((((0.386215468111418) + ((wt) * (-0.230516217152034))) + ((CASE WHEN (((char_cyl) = ('cyl6')) =  'TRUE') THEN (1.0) WHEN (((char_cyl) = ('cyl6')) =  'FALSE') THEN (0.0) END) * (0.332336511639638))) + ((CASE WHEN (((char_cyl) = ('cyl8')) =  'TRUE') THEN (1.0) WHEN (((char_cyl) = ('cyl8')) =  'FALSE') THEN (0.0) END) * (0.646203930513815))) * ((((0.386215468111418) + ((wt) * (-0.230516217152034))) + ((CASE WHEN (((char_cyl) = ('cyl6')) =  'TRUE') THEN (1.0) WHEN (((char_cyl) = ('cyl6')) =  'FALSE') THEN (0.0) END) * (0.332336511639638))) + ((CASE WHEN (((char_cyl) = ('cyl8')) =  'TRUE') THEN (1.0) WHEN (((char_cyl) = ('cyl8')) =  'FALSE') THEN (0.0) END) * (0.646203930513815))) * (6.63799055122669))) + (6.63799055122669))

Prediction intervals also works in the tidypredict_to_column(), just set the add_interval argument to TRUE.

df %>%
head(10)
##     mpg    wt char_cyl am      fit    upper    lower
## 1  21.0 2.620     cyl6  1 21.72241 27.41716 16.02765
## 2  21.0 2.875     cyl6  1 21.00014 26.65467 15.34560
## 3  22.8 2.320     cyl4  1 26.83929 32.35180 21.32678
## 4  21.4 3.215     cyl6  0 19.03711 24.68113 13.39309
## 5  18.7 3.440     cyl8  0 16.54108 22.07276 11.00940
## 6  18.1 3.460     cyl6  0 18.34317 24.01030 12.67603
## 7  14.3 3.570     cyl8  0 16.17286 21.67635 10.66938
## 8  24.4 3.190     cyl4  0 23.37507 29.06408 17.68606
## 9  22.8 3.150     cyl4  0 23.48837 29.16231 17.81443
## 10 19.2 3.440     cyl6  0 18.39981 24.06411 12.73552

## Under the hood

The parser reads several parts of the lm object to tabulate all of the needed variables. One entry per coefficient is added to the final table, those entries will have the results of qr.solve() already operated and placed in the correct column, they will have a qr_ prefix. There will be one qr_ column per coefficient.

Other variables are added at the end. Some variables are not required for every parsed model. For example, offset is listed because it’s part of the formula (call) of the model, if there were no offset in a given model, that line would not exist.

parse_model(model)
## # A tibble: 10 x 10
##    labels estima… type  fiel… fiel…    qr_1    qr_2     qr_3    qr_4 vals
##    <chr>    <dbl> <chr> <chr> <chr>   <dbl>   <dbl>    <dbl>   <dbl> <chr>
##  1 (Inte…   32.4  term  <NA>  <NA>  - 0.177 - 0.591 - 0.126    0.386 <NA>
##  2 wt      - 2.83 term  <NA>  {{:}}   0       0.184   0.0101 - 0.231 <NA>
##  3 char_…  - 4.27 term  cyl6  <NA>    0       0       0.428    0.332 <NA>
##  4 char_…  - 6.13 term  cyl8  <NA>    0       0       0        0.646 <NA>
##  5 labels    0    vari… char… wt     NA      NA      NA       NA     <NA>
##  6 model    NA    vari… <NA>  <NA>   NA      NA      NA       NA     lm
##  7 versi…   NA    vari… <NA>  <NA>   NA      NA      NA       NA     1.0
##  8 resid…   NA    vari… <NA>  <NA>   NA      NA      NA       NA     28
##  9 sigma2   NA    vari… <NA>  <NA>   NA      NA      NA       NA     6.63…
## 10 offset   NA    vari… <NA>  <NA>   NA      NA      NA       NA     am

The output from parse_model() is transformed into a dplyr, a.k.a Tidy Eval, formula. All categorical variables are operated using if_else().

tidypredict_fit(model)
## ((((32.4105336886021) + ((wt) * (-2.83243330448326))) + ((ifelse((char_cyl) ==
##     ("cyl6"), 1, 0)) * (-4.26714873091281))) + ((ifelse((char_cyl) ==
##     ("cyl8"), 1, 0)) * (-6.12588309683682))) + (am)

A function to put together the Tidy Eval interval formula is also supported

tidypredict_interval(model)
## (2.04840714179524) * sqrt((((((-0.176776695296637) * (-0.176776695296637) *
##     (6.63799055122669)) + (((-0.590557271637747) + ((wt) * (0.183559646169165))) *
##     ((-0.590557271637747) + ((wt) * (0.183559646169165))) * (6.63799055122669))) +
##     ((((-0.126215672528828) + ((wt) * (0.0101118696567173))) +
##         ((ifelse((char_cyl) == ("cyl6"), 1, 0)) * (0.428266330860589))) *
##         (((-0.126215672528828) + ((wt) * (0.0101118696567173))) +
##             ((ifelse((char_cyl) == ("cyl6"), 1, 0)) * (0.428266330860589))) *
##         (6.63799055122669))) + (((((0.386215468111418) + ((wt) *
##     (-0.230516217152034))) + ((ifelse((char_cyl) == ("cyl6"),
##     1, 0)) * (0.332336511639638))) + ((ifelse((char_cyl) == ("cyl8"),
##     1, 0)) * (0.646203930513815))) * ((((0.386215468111418) +
##     ((wt) * (-0.230516217152034))) + ((ifelse((char_cyl) == ("cyl6"),
##     1, 0)) * (0.332336511639638))) + ((ifelse((char_cyl) == ("cyl8"),
##     1, 0)) * (0.646203930513815))) * (6.63799055122669))) + (6.63799055122669))

From there, the Tidy Eval formula can be used anywhere where it can be operated. tidypredict provides three paths:

• Use directly inside dplyr, mutate(df, !! tidypredict_fit(model))
• Use tidypredict_to_column(model) to a piped command set
• Use tidypredict_to_sql(model) to retrieve the SQL statement

The same applies to the prediction interval functions.

## How it performs

Testing the tidypredict results is easy. The tidypredict_test() function automatically uses the lm model object’s data frame, to compare tidypredict_fit(), and tidypredict_interval() to the results given by predict()

tidypredict_test(model)
## tidypredict test results
## Difference threshold: 1e-12
##
##  All results are within the difference threshold

To run with prediction intervals set the include_intervals argument to TRUE

tidypredict_test(model, include_intervals = TRUE)
## tidypredict test results
## Difference threshold: 1e-12
##
##  All results are within the difference threshold