---
title: "Regression in rigr"
author: "Taylor Okonek, Brian D. Williamson, Yiqun T. Chen, and Amy D. Willis"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Regression in rigr}
  %\VignetteEngine{knitr::rmarkdown}
  \usepackage[utf8]{inputenc}
---

In the `rigr` package, we have set out to make regression and analysis easier by

1. allowing you to specify different types of regression from one function; 
2. automatically computing confidence intervals and p-values using robust standard errors; 
3. displaying output in a more intuitive fashion than base R; and
4. allowing you to specify multiple-partial F-tests.

This capability is implemented in the function `regress()`. The basic arguments to this function are

* `fnctl`: the functional
* `formula`: the formula for the linear model
* `data`: the data to use for the model

## `fnctl`: the functional

We use the concept of a *functional* to handle our first goal: allowing you to specify different types of regression models using a single function. A functional takes a *function* as its argument and returns a number. The most common example of a functional in regression is the *mean*. The allowed functionals to `regress()` are

|Functional | Type of Regression | base R command |
|---|:---|:---|
|`"mean"` | Linear Regression | `lm()` |
|`"geometric mean"` | Linear Regression on logarithmically transformed Y | `lm()`, with Y log-transformed |
|`"odds"` | Logistic Regression | `glm(family = binomial)` |
|`"rate"` | Poisson Regression | `glm(family = poisson)` |
| `"hazard"` | Proportional Hazards Regression | `coxph()` |

## `formula` and `data`

The *formula* to `regress()` is the same as a formula given to `lm()` or `glm()`, but with additional optional functionality to enable sophisticated analyses (multiple partial F-tests) with fewer headaches. 

<!-- To allow users to specify multiple partial F-tests, we have added a special function - `U()` - which can be added to the formula. `regress()` will compute an F statistic for all variables included inside the `U()` specification. -->

The *data* argument is exactly the same as that in `lm()` or any of the other regression commands.

# Linear Regression

As a first example, we run a linear regression of atrophy (a measure of global brain activity) on age, sex and race, from the `mri` data. This dataset is included in the `rigr` package; see its documentation by running `?mri`.

```{r}
## Preparing our R session
library(rigr)
data(mri)
regress("mean", atrophy ~ age + sex + race, data = mri)
```

Notice that by default robust standard error estimates are returned in addition to the naive estimates. The robust estimates are also used to perform inference. Thus, the confidence intervals, statistics, and p-values use these estimates of the standard error.

F-statistics are also displayed by default, including the multiple partial F-tests for the levels of a multi-level category (such as `race`) as well as the overall F-test for the variable.


# Generalized Linear Regression

## Logistic Regression

We can also run generalized linear regression using `regress()`. For example, to model the odds of having diabetes for males compared to females, we could run a logistic regression as follows:

```{r}
regress("odds", diabetes ~ sex, data = mri)
```

In all of the generalized linear regression output we see two tables. The `Raw Model` table displays estimated coefficients (and their standard errors) on the log-odds scale. The `Transformed Model` table exponentiates the estimated coefficients and their confidence intervals so that the estimated parameters can be interpreted on the odds scale. 

Note that the only possible link function in `regress` with `fnctl = odds"` is the logit link. Similarly, the only possible link function in `regress` with `fnctl = "rate"` is the log link.

## Poisson Regression

The next functional that `regress` supports is `"rate"`, for use in Poisson regression. To regress `yrsquit` on `age`, we would run:

```{r}
regress("rate", yrsquit ~ age, data = mri)
```

Note that again we have two tables of output, denoted by `Raw Model` and `Transformed Model`, with `Transformed Model` displaying exponentiated estimated coefficients. 

# Proportional Hazards Regression 

The final functional that `regress` supports is `"hazard"`, for use in proportional hazards regression. To regress `age` on the death status (note that we need to create a `Surv` object first), we would run:

```{r}
library(survival)
regress("hazard", Surv(obstime, death)~age, data=mri)
```

Similar to the Poisson regression case, we have two tables of output, denoted by `Raw Model` and `Transformed Model`, with `Transformed Model` displaying exponentiated estimated coefficients (i.e., on the hazard scale). 

# Regression on the Geometric Mean

Most often in linear regression we are interested in modeling the mean of the response variable. However, we are sometimes interested in modeling the mean of the log-transformed response variable, which allows us to make statements about the geometric mean of the response. In `regress()`, we can use the `"geometric mean"` functional to fit this model. Regression on the geometric mean of the `packyrs` variable in the `mri` dataset can be performed as follows:

```{r}
regress("geometric mean", packyrs ~ age, data = mri)
```

It should be noted that many of the `packyrs` observations are zero, but the geometric mean of data including an observation of zero is zero... regardless of how many non-zeros were also observed. Therefore, by default, zeroes in the outcome variable are replaced by a value equal to one-half the lowest nonzero value in the outcome variable. This is based on the idea that the lowest observed value could be a proxy for the lower limit of detection. If you wish to specify a different value with which to replace zeroes, you may do say using the `replaceZeroes` argument.

```{r}
regress("geometric mean", packyrs ~ age, data = mri, replaceZeroes = 1)
```

In the output from regress using the geometric mean functional, we see a table for the `Raw Model` and the `Transformed Model`. The `e(Est)`, `e(95%L)`, and `e(95%H)` columns in the `Transformed Model` table correspond to exponentiated values from the `Raw Model` - you'll notice that $e^{5.119} \approx 167.2.$

# Re-parameterizations of a Variable

There are two special functions in `rigr` which allow us to re-parameterize variables:

* `dummy`: create dummy variables
* `polynomial`: create a polynomial

Both of these functions may be used in a `regress()` call, and will additionally give a multiple partial F-test of the entire variable automatically.

## Specifying the reference group with `dummy`

The `dummy` function is useful for specifying the reference group that you wish to use with categorical variables. Below we show an example of using the reference group "Female" vs. the reference group "Male" in a regression on `sex`. 

```{r}
regress("mean", atrophy ~ dummy(sex, reference = "Male"), data = mri)
regress("mean", atrophy ~ dummy(sex, reference = "Female"), data = mri)
```

Notice that below the coefficients table in the output, the reference category is reported.

## Polynomial regression

You can fit higher-order polynomials using `polynomial`: 

```{r}
regress("mean", atrophy ~ polynomial(age, degree = 2), data = mri)
```

Note that all polynomials less than or equal to the degree specified are included in the model, and that the variables in the polynomial specification are mean-centered by default. You can change the centering using the `center` parameter in the polynomial function, an example of which is as follows.

```{r}
regress("mean", atrophy ~ polynomial(age, degree = 2, center = 65), data = mri)
```

# User-specified multiple partial F-tests

You can also perform multiple partial F-tests using formulas and the `U` function. This is useful when want to test a subset of variables all at once in your regression. For example, to test whether both the variables `packyrs` and `yrsquit` are associated with `atrophy` in a model with age as a predictor, we can run

```{r}
regress("mean", atrophy ~ age + sex + U("Smoking variables" = ~packyrs + yrsquit), data = mri)
```

`"Smoking variables"` is what we name the group of variables `packyrs` and `yrsquit`. The overall F statistic and p-value associated with the inclusion of these two smoking variables variables in the model are 4.37 and 0.0130, respectively.

# Testing contrasts: hypotheses about linear combinations of regression coefficients

You may be interested in testing a null hypothesis about a linear combination of coefficients in our regression model. For example, to investigate "Is the mean atrophy for a female subject equal to the mean atrophy for a male subject who is 10 years younger?", our hypothesis test involves both the coefficients on age and sex. We can test this hypothesis using the `lincom` function. First, we need to fit a linear model of age and sex on atrophy:

```{r}
mod_rigr <- regress("mean", atrophy ~ age + sex, data = mri)
mod_rigr
```

We then create a vector giving the linear combination of the coefficient that we hypothesized to be zero, and perform the test using `lincom`. The elements in `mod_combo` correspond to `Intercept`, `age`, and `sexMale`, because this was their order in the coefficient table shown above. 

```{r}
mod_combo <- c(0, -10, 1)
lincom(mod_rigr, mod_combo)
```

Note that the standard errors returned by default are robust, as are the associated confidence intervals and p-values.

We could also test the null hypothesis that the mean difference in atrophy between these two groups (females, and males 10 years younger) is equal to -1 as follows:

```{r}
lincom(mod_rigr, mod_combo, null.hypoth = -1)
```

