22-linear-models.qmd

# Linear models

## Measurement error models

* Another major application of linear models comes from measurement errors models.

* In these applications, it is common to have a non-random covariate, such as time, and randomness is introduced from measurement error rather than sampling or natural variability.

### Example: falling object

* Use **dslabs** function `rfalling_object` generates measurements of position taken from a falling object:

```{r, include=FALSE, warning=FALSE, cache=FALSE}
library(tidyverse)
library(broom)
library(dslabs)
falling_object <- rfalling_object()
```

* Here is the data:

```{r gravity}
falling_object |> 
  ggplot(aes(time, observed_distance)) + 
  geom_point() +
  ylab("Distance in meters") + 
  xlab("Time in seconds")
```

* The data seems to imply the trajectory follows a parabola, which we can write like this:

$$
f(x) = \beta_0 + \beta_1 x + \beta_2 x^2
$$

* But data does not fall exactly on a parabola. This most likely due to measurement error.

$$
Y_i = \beta_0 + \beta_1 x_i + \beta_2 x_i^2 + \varepsilon_i, i=1,\dots,n 
$$

* $Y_i$ representing distance in meters

* $x_i$ representing time in seconds

* $\varepsilon$ accounting for measurement error. 

* The measurement error is assumed to be random, independent from each other, and having the same distribution for each $i$. We also assume that there is no bias, which means the expected value $\mbox{E}[\varepsilon] = 0$.

Is this a liner model?


* Note that LSE calculations do not require the errors to be approximately normal so we can use `lm` to find the $\beta$s:

```{r}
library(broom)
fit <- falling_object |> 
  mutate(time_sq = time^2) |> 
  lm(observed_distance~time+time_sq, data = _)
tidy(fit)
```

Let's check if the estimated parabola fits the data. The **broom** function `augment` lets us do this easily:

```{r falling-object-fit}
augment(fit) |> 
  ggplot() +
  geom_point(aes(time, observed_distance)) + 
  geom_line(aes(time, .fitted), col = "blue")
```

* We actually know that the equation for the trajectory of a falling object is:

$$
d(t) = h_0 + v_0 t -  0.5 \times 9.8 \, t^2
$$

with $h_0$ and $v_0$ the starting height and velocity, respectively.

* These are consistent with the parameter estimates:

```{r}
tidy(fit, conf.int = TRUE)
```



## Treatment effect models {#sec-treatment-effect-models}

* Linear models can also be used to quantify treatment effects in randomized and controlled experiments. 

* One of the first applications was in agriculture,In fact the use of $Y$ for the outcome in Statistics, is due to the mathematical theory being developed for crop _yield_ as the outcome.  

* Since, the same ideas have been applied in other areas, such as randomized trials and _A/B testing_


* Furthermore, the use of these models has been extended to observational studies to adjust for _factors_ such as age, sex, and smoking status. 


## Mice weights

We are going to examine data from an experiment to see if a high fat diet increases weight.

```{r}
library(dslabs)
table(mice_weights$diet)
```

* A boxplot shows that the high fat diet mice are, on average, heavier.

```{r weight-by-diet-boxplots}
with(mice_weights, boxplot(body_weight ~ diet))
```

* But, given that we divided the mice at random, is it possible the observed difference is simply due to chance? 


## Comparing group means

The sample averages for the two groups, high-fat and chow diets, are different:

```{r}
library(tidyverse)
mice_weights |> group_by(diet) |> summarize(average = mean(body_weight))
```


Is this difference due to chance? 


## Hypothesis testing review

* Denote with $\mu_1$ and $\sigma_1$ the high-fat diet population average and standard deviation for weight. 

* Define $\mu_0$ and $\sigma_0$ similarly for the chow diet.

* Define $N_1$ and $N_0$ as the sample sizes, let's call them $\bar{X}_1$ and $\bar{X}_0$ as the sample averages, and $s_1$ and $s_0$ the sample standard deviations for the for the high-fat and chow diets, respectively.

* Because this is a random sample the central limit theorem tells us that the difference in averages 

$$
bar{X}_1 - \bar{X}_0
$$ 

follows a normal distribution with expected value $\mu_1-\mu_0$ and standard error $\sqrt{\frac{\sigma_1^2}{N_1} + \frac{\sigma_0^2}{N_0}}$. 

* If we define the null hypothesis as the high-fat diet having no effect, or $\mu_1 - \mu_0 = 0$, the the following summary statistic  


$$
t = \frac{\bar{X}_1 - \bar{X}_0}{\sqrt{\frac{s_1^2}{N_1} + \frac{s_0^2}{N_0}}}
$$

follows a standard normal distribution when the null hypothesis is true,  which implies we can easily compute the probability of observing a value as large as the one we did: 

```{r}
stats <- mice_weights |> group_by(diet) |> summarize(xbar = mean(body_weight), s = sd(body_weight), n = n()) 
t_stat <- with(stats, (xbar[2] - xbar[1])/sqrt(s[2]^2/n[2] + s[1]^2/n[1]))
t_stat
```

* Here $t$ is well over 3, so we don't really need to compute the p-value `1-pnorm(t_stat)` as we know it will be very small.

* Note that when $N$ is not large, then the CLT does not apply. However, if the outcome data, in this case weight, follows a normal distribution, then $t$ follows a t-distribution with $N_1+N_2-2$ degrees of freedom. 

* So the calculation of the p-value is the same except we use `1-pt(t_stat, with(stats, n[2]+n[1]-2)` to compute the p-value. 

* Because using differences in mean are so common in scientific studies, this _t-statistic_ is one of the most widely reported summaries. When use it in a hypothesis testing setting, it is referred to a performing a _t test_. 


:::{.callout-warning}
In the computation above we computed the probability of `t` being as large as what we observed. However, when we are equally interested in both directions, for example, either an increase or decrease in weight, then we need to compute the probability of `t` being _as extreme_ as what we observe. The formula simply changes to using the absolute value: `1 - pnorm(abs(t-test))` or `1-pt(t_stat, with(stats, n[2]+n[1]-2)`.
:::

## One factor design

* Although the t-test is useful for cases in which we only account for two treatments, it is common to have other variables affect our outcomes. 

* Linear models permit hypothesis testing in more general situations. 

* We start the description of the use linear models for estimating treatment effects by demonstrating how they can be used to perform t-tests.

* If we assume that the weight distributions for both chow and high-fat diets are normally distributed, we can write the following linear model to represent the data:

$$
Y_i = \beta_0 + \beta_1 x_i + \varepsilon_i
$$

* with $X_i$ 1 if the $i$-th mice was fed the high-fat diet and 0 otherwise and the errors $\varepsilon_i$ independent and normally distributed with expected value 0 and standard deviation $\sigma$. 


* Notice that now $\beta_0$ represents the population average height of the mice on the chow diet and $\beta_0 + \beta_1$ represents the population average for the weight of the mice on the high-fat diet. 

* A nice feature of this model is that $\beta_1$ represents the _treatment effect_ of receiving the high-fat diet. 

* If the null hypothesis that the high-fat diet has no effect can be quantified as $\beta_1 = 0$. 

* We can then estimate $\beta_1$ and answer the question of weather or not the observed difference is real by computing the estimates being as large as it was under the null.

* A powerful characteristics of linear models is that we can can estimate the parameters $\beta$s and their standard errors with the same LSE machinery:

```{r}
fit <- lm(body_weight ~ diet, data = mice_weights)
```

* Because `diet` is a factor with two entries, the `lm` function knows to fit the model above with a $x_i$ a indicator variable. The `summary` function shows us the resulting estimate, standard error, and p-value:

```{r}
coefficients(summary(fit))
```

or using `broom` we can write:

```{r}
tidy(fit, conf.int = TRUE) |> filter(term == "diethf")
```

* The `statistic` computed here is the estimate divided by its standard error: 

$$
\hat{\beta}_1 / \hat{\mbox{SE}}(\hat{\beta}_1)
$$

* In the case for the simple one-factor model, we can show that this statistic is almost equivalent to the t-test. 


```{r}
c(coefficients(summary(fit))[2,3], t_stat)
```


* One minor difference is that the linear model does not assume a different standard deviation for each population.


:::{.callout-note}
In the linear model description provided here we assumed $\varepsilon$ follows a normal distribution. This assumption permits us to show that the statistics formed by dividing estimates by their estimated standard errors follow t-distribution, which in turn permits us to estimate p-values or confidence intervals. However, note that we do not need this assumption to compute the expected value and standard error of the least squared estimates. Furthermore, if the number of observations in large enough, then the central limit theorem applies and we can obtain p-values and confidence intervals even without the normal distribution assumption.
:::

## Two factor designs

* Note that this experiment included male and female mice, and male mice are known to be heavier. '

* Note the residuals depend on the sex variable:

```{r lm-residual-boxplots}
boxplot(fit$residuals ~ mice_weights$sex)
```

* This misspecification can have real implications since if more male mice received the high-fat diet, then this could explain the increase.

* Or if less received it, then we might underestimate the diet effect. Sex might be a confounder. Our model can certainly be improved. 

* From examining the data:

```{r weight-by-sex-diet-boxplots}
mice_weights |> ggplot(aes(diet, log2(body_weight), fill = sex)) + geom_boxplot()
```

we see that there diet effect is observed for both sexes and that males are heavier than females. 

* Although not nearly as obvious, it also appears the diet effect is stronger in males. A linear models that permits a different expected value four groups, 1) female on chow diet, 2) females on high-fat diet, 3) male on chow diet, and 4)males on high-fat diet, 

$$
Y_i = \beta_1 x_{i,1} + \beta_2 x_{i,2}  + \beta_3 x_{i,3}  + \beta_4 x_{i,4}  + \varepsilon_i
$$

with the $x_i$s indicator variables for each of the four groups. 

* However, with this representation, none of the $\beta$s represent the effect of interest: the diet effect. 

* Furthermore, we now are accounting for the possibility that the diet effect is different for males and females have a different, and can test that hypothesis as well. 


* A powerful feature of linear models is that we can rewrite the model so that we still have a different expected value for each group, but the parameters represent effects we are interested. So, for example, in the representation

$$
Y_i = \beta_0 + \beta_1 x_{i,1}  + \beta_2 x_{i,2}  + \beta_3 x_{i,1} x_{i,2}  + \varepsilon_i
$$

with $x_{i,1}$ and indicator that is one if you have the treatment and $x_{i,2}$ an indicator that is one if you are male, the $\beta_1$ can be interpreted as the treatment effect for females, $\beta_2$ as the difference between males and females, and $\beta_3$ the added treatment effect for males. 

* This last effect is referred to as an _interaction_ effect. 

* The $\beta_0$ is consider the baseline value which is the average weight of females on the chow diet. 

* Statistical textbooks describes several other ways in which the model can be rewritten to obtain other types of interpretations. For example, we might want $\beta_2$ to represent the average treatment effect between females and males, rather that the female treatment effects. This is achieved by defining what _contrasts_ we are interested. 

* In R we can specific this model using the following

```{r}
fit <- lm(body_weight ~ diet*sex, data = mice_weights)
```

* The `*` implies that the term that multiplies $x_{i,1}$ and $x_{i,2}$ should be included, along with the  $x_{i,1}$ and $x_{i,2}$ terms. 

```{r}
tidy(fit, conf.int = TRUE) |> filter(!str_detect(term, "Intercept"))
```

* Note that the male effect is larger that the diet effect, and the diet effect is statistically significant for both sexes, with the males having a higher effect by between 1 and 4.5 grams.


* A common approach applied when more than one factor is thought to affect the measurement is to simply include an additive effect for each factor like this:

$$
Y_i = \beta_0 + \beta_1 x_{i,1}  + \beta_2 x_{i,2}   + \varepsilon_i
$$

* In this model, the $\beta_1$ is a general diet effect that applies regardless of sex. In R we use the following code using a `+` instead of `*`:

```{r}
fit <- lm(body_weight ~ diet + sex, data = mice_weights)
```

* Because their a strong interaction effect, a diagnostic plots shows that the residuals are biased: the average negative for females on the diet and positive for the males on the diet, rather than 0.

```{r lm-diagnostic-plot}
plot(fit, which = 1)
```

* Scientific studies, particularly within epidemiology and social sciences, frequently omit interaction terms from models due to the high number of variables. 

* Adding interactions necessitates numerous parameters, which, in extreme cases, may prevent the model from fitting. However, this approach assumes that the interaction terms are zero, which, if incorrect, can skew the results' interpretation. 

* Conversely, when this assumption is valid, models excluding interactions are simpler to interpret as parameters are typically viewed as the extent to which the outcome increases with the assigned treatment.

:::{.callout-tip}
Linear models are very flexible and can be applied in many contexts. For example, we can include many more factors than 2. We have just scratched the surface of how linear models can be used to estimate treatment effects. We highly recommend learning more about this through linear model textbooks and R manuals on using the `lm`, `contrasts`, and `model.matrix` functions.
:::


## Analysis of variance


* Often we have variables of interest that have more than one level. 

* For example, we might have tested a third diet on the mice. 

* In statistics textbooks these variables are referred to as _factor_. 

* In these cases it is common to want to know rather than the effect of each levels of the factor, a more general quantification regarding the variability across the levels. 

* Analysis of variances or ANOVA does just this. 

* The summary used to quantify the variability of a factor is the mean squared error of the estimated effects of each level.

* As an example, consider that the mice in our dataset are actually from several generations: 

```{r}
table(mice_weights$gen)
```

* We can fit a linear model that fits an effect for each of these generations along with diet and sex model previously fit:

```{r}
fit <- lm(body_weight ~ diet * sex + gen,  data = mice_weights)
```

We can then perform an analysis of variance with the R `aov` function:

```{r}
summary(aov(fit))
```

This analysis shows that the largest variation is explained by sex and then diet. The generation factor explains very little variation in comparison and is not found to be statistically significant. 

:::{.callout-note}
In this book, we do not provide the details for how we compute this p-value. There are several books on analysis of variance and textbooks on linear models often include chapters on this topic. Those interested in learning more about these topics can consult these textbooks.
:::




## Exercises 

(@) Plot of co2 evels for the first 12 months of the `co2` dataset and notice it seems to follow a sin wave with frequency 1 cycle per month. This means that a measurement error model that might work is 

$$
y_i = \mu + A \sin(2\pi t_i / 12 + \phi) + \varepsilon_i
$$
with $t_i$ the month number of observation $i$. Is this a linear model for the parameters $mu$, $A$ and $\phi$?


(@) Using trigonometry we can show that we can rewrite this model as

$$
y_i = \beta_0 + \beta_1 \sin(2\pi t_i/12) + \beta_2 \cos(2\pi t_i/12) + \varepsilon_i
$$


(@) Find least square estimates for the $\beta$s using `lm`. Show a plot of $y_i$ versus $t_i$ with a curve on the same plot showing $\hat{Y}_i$ versus $t_i$.

```{r}
y <- as.vector(co2)[1:12]
tt <- seq_along(y)
plot(tt,y)
x1 <- sin(2*pi*tt/12)
x2 <- cos(2*pi*tt/12)
fit <- lm(y~x1+x2)
lines(tt, fit$fitted)
```

(@) Now fit a measurement error model to the entire `co2` dataset that includes a trend term that is a parabola as well as the sine wave model.

```{r}
y <- as.vector(co2)
tt <- seq_along(y)
x1 <- sin(2*pi*tt/12)
x2 <- cos(2*pi*tt/12)
fit <- lm(y~x1+x2+poly(tt,2))
plot(tt, y)
lines(tt, fit$fitted, col = 2)
summary(fit)
```

(@) Run diagnostic plots for the fitted model and describe the results.


(@) Generate a sample of size $N=50$ from an urn model with 50% blue beads:

```{r, eval=FALSE}
N <- 50 
p <- 0.5
x <- rbinom(N, 1, 0.5)
```

Then compute a p-value testing if $p=0.5$. Repeat this 10,000 times and report how often do we incorrectly is the p-value lower than 0.05? How often is it lower than 0.01?

(@) Make a histogram of the p-values you generated in exercise 1. Which of the following seems to be true:

  a. The p-values are all 0.05
  b. The p-values are normally distributed, CLT seems to hold.
  c. The p-values are uniformly distributed
  d. The p-values 


(@)  Generate a sample of size $N=50$ from an urn model with 52% blue beads:

```{r, eval=FALSE}
N <- 50 
p <- 0.52
x <- rbinom(N, 1, 0.5)
```

Then compute a p-value testing if $p=0.5$. Repeat this 10,000 times and report how often do we incorrectly is the p-value larger than 0.05? Note that you are computing 1 - power. 

(@) Repeat exercise for but for the following values:

```{r}
values <- expand.grid(N = c(25, 50, 100, 500, 1000), p = seq(0.51 ,0.75, 0.01))
```

Plot power as a function of $N$ with a different color curve for each value of `p`.


(@) Once you fit a model, the estimate of the standard error $\sigma$ can be obtained like this:

```{r}
#| eval: false

fit <- lm(body_weight ~ diet, data = mice_weights)
summary(fit)$sigma
```

Compute the estimate of $\sigma$ using the model that includes just diet and a model that accounts for sex. Are the estimates the same? If not, why not?


(@) One of the assumption of the linear model fit by `lm` is that the standard deviation of the errors $\varepsilon_i$ is equal for all $i$. This implies it does not depend on the expected value. Group the mice by their weight like this:

```{r}
breaks <- with(mice_weights, seq(min(body_weight), max(body_weight), 1))
dat <- mutate(mice_weights, group = cut(body_weight, breaks, include_lowest = TRUE))
```

Compute the average and standard deviation for groups having more than 10 observations and use data exploration to see if this assumption holds?

(@) The dataset also includes a variable indicating which litter the mice came from. Create a boxplot showing weights by litter. Use faceting to make separate plots for each diet and sex combination.

(@) Use a linear model to test for a litter effect. Account for sex and diet. Use ANOVA to compare the variability explained by litter to other factors.

(@) The `mouse_weights` data includes two other outcomes: bone density and percent fat. Make a boxplot showing bone density by sex and diet. Compare what the visualizations shows for the diet effect by sex.

(@) Fit a linear model and test for the diet effect on bone density separately for each sex. Note that the diet effect is statistically significant for females but not for males. Then fit the model to the entire dataset that includes diet, sex and their interaction. Note that the diet effect is significant, yet the interaction effect is not. Explain how this can happen? Hint: To fit a model to the entire dataset that fit a separate effect for males and females you can use the formula ` ~ sex + diet:sex`

(@) We previously talked about pollster bias. We used visualization to motivate the presence of such bias. Here we will give it a more rigorous treatment. We will consider two pollsters that conducted daily polls. We will look at national polls for the month before the election.

```{r, eval=FALSE}
polls <- polls_us_election_2016 |> 
  filter(pollster %in% c("Rasmussen Reports/Pulse Opinion Research",
                         "The Times-Picayune/Lucid") &
           enddate >= "2016-10-15" &
           state == "U.S.") |> 
  mutate(spread = rawpoll_clinton/100 - rawpoll_trump/100) 
```

We want to answer the question: is there pollsetr bias? Make a plot showing the spreads for each pollster.

(@) The data does seem to suggest there is a difference. However, these data are subject to variability. Perhaps the differences we observe are due to chance.

The urn model theory says nothing about pollster effect. Under the urn model, both pollsters have the same expected value: the election day difference, that we call $\mu$. Use a linear model to answer "is there an pollster effect?"