# Linear Regression

## 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

    library(dplyr)
    library(tidypredict)

    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') THEN (1.0) WHEN NOT(`char_cyl` = 'cyl6') THEN (0.0) END * -4.26714873091281) + (CASE WHEN (`char_cyl` = 'cyl8') THEN (1.0) WHEN NOT(`char_cyl` = 'cyl8') 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') THEN (1.0) WHEN NOT(`char_cyl` = 'cyl6') THEN (0.0) END * 0.428266330860589 * -0.126215672528828 + `wt` * 0.0101118696567173 + CASE WHEN (`char_cyl` = 'cyl6') THEN (1.0) WHEN NOT(`char_cyl` = 'cyl6') THEN (0.0) END * 0.428266330860589 * 6.63799055122669 + 0.386215468111418 + `wt` * -0.230516217152034 + CASE WHEN (`char_cyl` = 'cyl6') THEN (1.0) WHEN NOT(`char_cyl` = 'cyl6') THEN (0.0) END * 0.332336511639638 + CASE WHEN (`char_cyl` = 'cyl8') THEN (1.0) WHEN NOT(`char_cyl` = 'cyl8') THEN (0.0) END * 0.646203930513815 * 0.386215468111418 + `wt` * -0.230516217152034 + CASE WHEN (`char_cyl` = 'cyl6') THEN (1.0) WHEN NOT(`char_cyl` = 'cyl6') THEN (0.0) END * 0.332336511639638 + CASE WHEN (`char_cyl` = 'cyl8') THEN (1.0) WHEN NOT(`char_cyl` = 'cyl8') 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 %>%
      tidypredict_to_column(model, add_interval = TRUE) %>%
      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.

    pm <- parse_model(model)
    str(pm, 2)
    #> List of 2
    #>  $ general:List of 7
    #>   ..$ model   : chr "lm"
    #>   ..$ version : num 2
    #>   ..$ type    : chr "regression"
    #>   ..$ residual: int 28
    #>   ..$ sigma2  : num 6.64
    #>   ..$ offset  : symbol am
    #>   ..$ is_glm  : num 0
    #>  $ terms  :List of 4
    #>   ..$ :List of 5
    #>   ..$ :List of 5
    #>   ..$ :List of 5
    #>   ..$ :List of 5
    #>  - attr(*, "class")= chr [1:3] "parsed_model" "pm_regression" "list"

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

## parsnip

`tidypredict` also supports `lm()` model objects fitted via the `parsnip` package.

    library(parsnip)

    parsnip_model <- linear_reg() %>%
      set_engine("lm") %>%
      fit(mpg ~ wt + cyl, offset = am, data = mtcars)

    tidypredict_fit(parsnip_model)
    #> 39.6862614802529 + (wt * -3.19097213898374) + (cyl * -1.5077949682598)