IDS 702: Module 4.3

class: center, middle, inverse, title-slide

# IDS 702: Module 4.3
## Multilevel/hierarchical linear models (illustration I)
### Dr. Olanrewaju Michael Akande

---

## The radon analysis

There are 919 total observations in the data. The data is in the file `Radon.txt` on Sakai.

---
## The radon analysis

```r
Radon <- read.csv("data/Radon.txt", header = T,sep="")
Radon$floor <- factor(Radon$floor,levels=c(0,1),labels=c("Basement","First Floor"))
str(Radon)
```

```
## 'data.frame':	919 obs. of  9 variables:
##  $ radon      : num  2.2 2.2 2.9 1 3.1 2.5 1.5 1 0.7 1.2 ...
##  $ state      : chr  "MN" "MN" "MN" "MN" ...
##  $ log_radon  : num  0.788 0.788 1.065 0 1.131 ...
##  $ floor      : Factor w/ 2 levels "Basement","First Floor": 2 1 1 1 1 1 1 1 1 1 ...
##  $ countyname : chr  "AITKIN" "AITKIN" "AITKIN" "AITKIN" ...
##  $ countyID   : int  1 1 1 1 2 2 2 2 2 2 ...
##  $ fips       : int  27001 27001 27001 27001 27003 27003 27003 27003 27003 27003 ...
##  $ uranium    : num  0.502 0.502 0.502 0.502 0.429 ...
##  $ log_uranium: num  -0.689 -0.689 -0.689 -0.689 -0.847 ...
```

```r
head(Radon)
```

```
##   radon state log_radon       floor countyname countyID  fips  uranium
## 1   2.2    MN 0.7884574 First Floor     AITKIN        1 27001 0.502054
## 2   2.2    MN 0.7884574    Basement     AITKIN        1 27001 0.502054
## 3   2.9    MN 1.0647107    Basement     AITKIN        1 27001 0.502054
## 4   1.0    MN 0.0000000    Basement     AITKIN        1 27001 0.502054
## 5   3.1    MN 1.1314021    Basement      ANOKA        2 27003 0.428565
## 6   2.5    MN 0.9162907    Basement      ANOKA        2 27003 0.428565
##   log_uranium
## 1  -0.6890476
## 2  -0.6890476
## 3  -0.6890476
## 4  -0.6890476
## 5  -0.8473129
## 6  -0.8473129
```

```r
summary(Radon[,-c(2,7)])
```

```
##      radon          log_radon               floor      countyname       
##  Min.   : 0.000   Min.   :-2.3026   Basement   :766   Length:919        
##  1st Qu.: 1.900   1st Qu.: 0.6419   First Floor:153   Class :character  
##  Median : 3.600   Median : 1.2809                     Mode  :character  
##  Mean   : 4.768   Mean   : 1.2246                                       
##  3rd Qu.: 6.000   3rd Qu.: 1.7918                                       
##  Max.   :48.200   Max.   : 3.8754                                       
##     countyID        uranium        log_uranium      
##  Min.   : 1.00   Min.   :0.4140   Min.   :-0.88183  
##  1st Qu.:21.00   1st Qu.:0.6221   1st Qu.:-0.47467  
##  Median :44.00   Median :0.9080   Median :-0.09652  
##  Mean   :43.52   Mean   :0.9339   Mean   :-0.13171  
##  3rd Qu.:70.00   3rd Qu.:1.2011   3rd Qu.: 0.18324  
##  Max.   :85.00   Max.   :1.6956   Max.   : 0.52802
```

---
## The radon analysis

.midsmall[

```r
table(Radon$countyname) #we don't have enough data in some counties, so we should look to borrow information across counties.
```

```
## 
##            AITKIN             ANOKA            BECKER          BELTRAMI 
##                 4                52                 3                 7 
##            BENTON         BIG STONE        BLUE EARTH             BROWN 
##                 4                 3                14                 4 
##           CARLTON            CARVER              CASS          CHIPPEWA 
##                10                 6                 5                 4 
##           CHISAGO              CLAY        CLEARWATER              COOK 
##                 6                14                 4                 2 
##        COTTONWOOD         CROW WING            DAKOTA             DODGE 
##                 4                12                63                 3 
##           DOUGLAS         FARIBAULT          FILLMORE          FREEBORN 
##                 9                 6                 2                 9 
##           GOODHUE          HENNEPIN           HOUSTON           HUBBARD 
##                14               105                 6                 5 
##            ISANTI            ITASCA           JACKSON           KANABEC 
##                 3                11                 5                 4 
##         KANDIYOHI           KITTSON       KOOCHICHING     LAC QUI PARLE 
##                 4                 3                 7                 2 
##              LAKE LAKE OF THE WOODS          LE SUEUR           LINCOLN 
##                 9                 4                 5                 4 
##              LYON          MAHNOMEN          MARSHALL            MARTIN 
##                 8                 1                 9                 7 
##            MCLEOD            MEEKER        MILLE LACS          MORRISON 
##                13                 5                 2                 9 
##             MOWER            MURRAY          NICOLLET            NOBLES 
##                13                 1                 4                 3 
##            NORMAN           OLMSTED        OTTER TAIL        PENNINGTON 
##                 3                23                 8                 3 
##              PINE         PIPESTONE              POLK              POPE 
##                 6                 4                 4                 2 
##            RAMSEY           REDWOOD          RENVILLE              RICE 
##                32                 5                 3                11 
##              ROCK            ROSEAU             SCOTT         SHERBURNE 
##                 2                14                13                 8 
##            SIBLEY          ST LOUIS           STEARNS            STEELE 
##                 4               116                25                10 
##           STEVENS             SWIFT              TODD          TRAVERSE 
##                 2                 4                 3                 4 
##           WABASHA            WADENA            WASECA        WASHINGTON 
##                 7                 5                 4                46 
##          WATONWAN            WILKIN            WINONA            WRIGHT 
##                 3                 1                13                13 
##   YELLOW MEDICINE 
##                 2
```
]

---
## The radon analysis

The raw radon levels can only take on positive values.

```r
ggplot(Radon,aes(radon)) +
  geom_histogram(aes(y=..density..),color="black",linetype="dashed",
                 fill=rainbow(15),bins=15) + theme(legend.position="none") +
  geom_density(alpha=.25, fill="lightblue") + scale_fill_brewer(palette="Blues") +
  labs(title="Distribution of Radon Levels",y="Radon") + theme_classic() 
```

--
.block[Obviously very skewed.]

---
## The radon analysis

Let's look at `log_radon` instead.

```r
ggplot(Radon,aes(log_radon)) +
  geom_histogram(aes(y=..density..),color="black",linetype="dashed",
                 fill=rainbow(15),bins=15) + theme(legend.position="none") +
  geom_density(alpha=.25, fill="lightblue") + scale_fill_brewer(palette="Blues") +
  labs(title="Distribution of Log Radon Levels",y="Log Radon") + theme_classic() 
```

.block[Much better! Let's go with log radon for now.]

---
## The radon analysis

Are there any variations of radon levels by county? There are too many counties, so, let's do it for a random sample of counties.

```r
set.seed(1000)
sample_county <- sample(unique(Radon$countyname),25,replace=F)
ggplot(Radon[is.element(Radon$countyname,sample_county),],
       aes(x=countyname, y=log_radon, fill=countyname)) +
  geom_boxplot() +
  labs(title="Log radon levels by county",
       x="County",y="Log Radon") + theme_classic() + 
  theme(legend.position="none",axis.text.x = element_text(angle = 90))
```

---
## The radon analysis

.block[Looks like the levels vary by county. However, there are many counties with very little data.]

---
## The radon analysis

Let's focus on counties with at least 11 houses.

```r
sample_county <- which(table(Radon$countyID) > 10)
ggplot(Radon[is.element(Radon$countyID,sample_county),],
       aes(x=countyname, y=log_radon, fill=countyname)) +
  geom_boxplot() +
  labs(title="Log radon levels by county",
       x="County",y="Log Radon") + theme_classic() + 
  theme(legend.position="none",axis.text.x = element_text(angle = 90))
```

---
## The radon analysis

<div class="question">
What can you conclude from this plot?
</div>

---
## The radon analysis

Next, the relationship with `floor`, the only individual-level (different observation for each house) variable we have.

```r
ggplot(Radon,aes(x=floor, y=log_radon, fill=floor)) +
  geom_boxplot() + scale_fill_brewer(palette="Greens") +
  labs(title="Log radon vs floor", x="Lowest living area of each house",y="Log Radon") + 
  theme_classic() + theme(legend.position="none")
```

.block[Looks like radon levels are higher for houses with the basement as the lowest living area.]

---
## The radon analysis

Let's look at the same relationship for a random sample of counties.

```r
sample_county <- sample(unique(Radon$countyname),8,replace=F)
ggplot(Radon[is.element(Radon$countyname,sample_county),],
       aes(x=floor, y=log_radon, fill=floor)) +
  geom_boxplot() +
  scale_fill_brewer(palette="Greens") +
  labs(title="Log radon vs floor by county",
       x="Lowest living area of each house",y="Log Radon") + 
  theme_classic() + theme(legend.position="none") +
  facet_wrap( ~ countyname,ncol=4)
```

---
## The radon analysis

Again, not enough data for some counties.

---
## The radon analysis

Let's focus on counties with at least 16 houses.

```r
sample_county <- which(table(Radon$countyID) > 15)
ggplot(Radon[is.element(Radon$countyID,sample_county),],
       aes(x=floor, y=log_radon, fill=floor)) +
  geom_boxplot() +
  scale_fill_brewer(palette="Greens") +
  labs(title="Log radon vs floor by county",
       x="Lowest living area of each house",y="Log Radon") + 
  theme_classic() + theme(legend.position="none") +
  facet_wrap( ~ countyname,ncol=4)
```

---
## The radon analysis

.block[Even though the overall direction is the same, it looks like the actual differences between floor = 0 and floor = 1 differs for some counties.]

---
## The radon analysis

- Let's start by only focusing on `floor`.

- We will try a  varying-slope, varying-intercept linear model.

- Let `$y_{ij}$` and `$x_{1ij}$` be the log radon level and indicator variable `floor` respectively for house `$i$` in county `$j$`.

- Mathematically, we have
.block[
.small[
$$
`\begin{split}
y_{ij} & = (\beta_{0} + \gamma_{0j}) + (\beta_1 + \gamma_{1j}) x_{1ij} + \epsilon_{ij}; \ \ \ i = 1, \ldots, n_j; \ \ \ j = 1, \ldots, 85\\
\epsilon_{ij} & \sim N(0, \sigma^2) \\
(\gamma_{0j},\gamma_{1j}) & \sim N_2(\boldsymbol{0}, \Sigma).
\end{split}`
$$
]
]

- Alternative representation:
.block[
.small[
$$
`\begin{split}
\text{log(radon}_{ij}\text{)} & = (\beta_{0} + \gamma_{0j}) + (\beta_1 + \gamma_{1j}) \text{ floor}_{ij} + \epsilon_{ij}; \ \ \ i = 1, \ldots, n_j; \ \ \ j = 1, \ldots, 85 \\
\epsilon_{ij} & \sim N(0, \sigma^2) \\
(\gamma_{0j},\gamma_{1j}) & \sim N_2(\boldsymbol{0}, \Sigma).
\end{split}`
$$
]
]

---
## The radon analysis

- We skipped this before but `$\Sigma$` actually takes the form
.block[
.small[
$$
\Sigma =
`\begin{bmatrix}
\tau_0^2 & \rho \tau_0\tau_1  \\
\rho \tau_0\tau_1 & \tau_1^2 \\
\end{bmatrix}`
$$
]
]

where
  + `$\tau_0^2$` describes the across county variation attributed to the random/varying intercept,
  
  + `$\tau_1^2$` describes the across county variation attributed to the random/varying slope (that is, floor), and
  
  + `$\rho$` describes the correlation between `$\gamma_{0j}$` and `$\gamma_{1j}$`.

---
## The radon analysis

In R, we have

```r
Model1 <- lmer(log_radon ~ floor + (floor | countyname), data = Radon) 
summary(Model1)
```

```
## Linear mixed model fit by REML ['lmerMod']
## Formula: log_radon ~ floor + (floor | countyname)
##    Data: Radon
## 
## REML criterion at convergence: 2168.3
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -4.4044 -0.6224  0.0138  0.6123  3.5682 
## 
## Random effects:
##  Groups     Name             Variance Std.Dev. Corr 
##  countyname (Intercept)      0.1216   0.3487        
##             floorFirst Floor 0.1181   0.3436   -0.34
##  Residual                    0.5567   0.7462        
## Number of obs: 919, groups:  countyname, 85
## 
## Fixed effects:
##                  Estimate Std. Error t value
## (Intercept)       1.46277    0.05387  27.155
## floorFirst Floor -0.68110    0.08758  -7.777
## 
## Correlation of Fixed Effects:
##             (Intr)
## florFrstFlr -0.381
```

---
## Interpretation of fixed effects

- Intuitively, we have an overall "average" regression line for all houses across all counties in Minnesota which has slope -0.68 and intercept 1.46.

- That is, the general estimated line for any of the houses in Minnesota is: 
.block[
.small[
$$ \widehat{\text{log(radon}_{i}\text{)}} = 1.46 - 0.68 \times \textrm{floor}_i $$
]
]

- For .hlight[any house in Minnesota with a basement as the lowest living area, the baseline radon level is] `$e^{1.46} = 4.31$`.

- Then, for any house in Minnesota, .hlight[having a first floor as the lowest living area, instead of a basement], reduces the radon level by a multiplicative effect of `$e^{-0.68} = 0.51$`, that is, about a 49% reduction.

- However, if the house is in Dakota county for example, we also need to add on the random intercepts and slopes for that county.

---
## Interpretation of fixed effects

- For Dakota county, we have
  
  ```r
  (ranef(Model1)$countyname)["DAKOTA",]
  ```
  
  ```
  ##        (Intercept) floorFirst Floor
  ## DAKOTA  -0.1099052      -0.08786805
  ```

so that the estimated regression line for Dakota county is actually
.block[
.small[
$$ \widehat{\text{log(radon}_{i}\text{)}} = (1.46 - 0.11) + (-0.68-0.09) \times \textrm{floor}_i = 1.35 - 0.77 \times \textrm{floor}_i $$
]
]

- Thus, for any house in Dakota county in Minnesota with a basement as the lowest living area, the baseline radon level is actually `$e^{1.35} = 3.86$`, which is .hlight[lower than the overall state wide average].

- And for any house in Dakota county in Minnesota, having the first floor be the lowest living area then reduces the radon level by a multiplicative effect of `$e^{-0.77} = 0.46$`, that is about a 54% reduction, .hlight[more than the overall state wide effect].

---
## The radon analysis

Again,

```r
summary(Model1)
```

---
## Interpretation of random effects

- The estimated standard error `$\hat{\sigma} = 0.75$` describes the within-county or remaining unexplained variation.

- The estimated `$\hat{\tau_0} = 0.35$` describes the across-county variation attributed to the random intercept.

- The estimated `$\hat{\tau_1} = 0.34$` describes the across-county variation attributed to the random slope (the predictor, floor).

- Those two sources of county variation are actually quite similar.

- The estimated correlation between `$\gamma_{0j}$` and `$\gamma_{1j}$` is `$\hat{\rho} = -0.34$`.

- You can visualize the random effects by typing `dotplot(ranef(Model1, condVar=TRUE))$countyname` in R.

- So many counties! So, you will need to zoom out on your computer.

---
## Interpretation of random effects

---

class: center, middle

# What's next?

### Move on to the readings for the next module!