Drawing inference 🤔

# Drawing inference <br> 🤔

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<div class="my-footer">
<span>
by Dr. Lucy D'Agostino McGowan
</span>
</div>

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## <i class="fas fa-laptop"></i> `Porsche Price (3)`

- Go to RStudio Cloud and open `Porsche Price (3)`

---
class: middle

# in·fer·ence
a conclusion reached on the basis of evidence and reasoning.

---
# Inference

* so far we've only been able to make claims about our sample
--

* for example, we've just been describing `\(\hat{\beta}_1\)`, the _estimated_ slope of the relationship between `\(x\)` and `\(y\)`.
--

* what if we want to extend these claims to the population?

---

# Sparrow data

_So far, we've been looking at a sample of 116 sparrows from Kent Island._

![](06-drawing-inference_files/figure-html/unnamed-chunk-2-1.png)

---

## Sparrows

.question[
What if this were the true population, and the sample that we saw was just related _by chance_?
]

![](06-drawing-inference_files/figure-html/unnamed-chunk-3-1.png)

---

## Sparrows

![](06-drawing-inference_files/figure-html/unnamed-chunk-4-1.png)

---

## Sparrows

![](06-drawing-inference_files/figure-html/unnamed-chunk-5-1.png)

* Does the slope in the **population** differ from 0?

---

## Sparrows

![](06-drawing-inference_files/figure-html/unnamed-chunk-6-1.png)

* Does `\(\beta_1\)` differ from 0?
--

* _notice the lack of a hat!_
---

## Sparrows

![](06-drawing-inference_files/figure-html/unnamed-chunk-7-1.png)

* **null hypothesis** `\(H_0: \beta_1 = 0\)` 
* **alternative hypothesis** `\(H_A: \beta_1 \ne 0\)`

---

## Sparrows

.question[
How can we quantify how much we'd expect the slope to differ from one random sample to another?
]

![](06-drawing-inference_files/figure-html/unnamed-chunk-8-1.png)

---

## Sparrows

.question[
How can we quantify how much we'd expect the slope to differ from one random sample to another?
]

![](06-drawing-inference_files/figure-html/unnamed-chunk-9-1.png)

* We need a measure of **uncertainty**

---

## Sparrows

.question[
How can we quantify how much we'd expect the slope to differ from one random sample to another?
]

![](06-drawing-inference_files/figure-html/unnamed-chunk-10-1.png)

* How about the **standard error** of `\(\hat{\beta}_1\)`?

---

## Sparrows

.question[
How can we quantify how much we'd expect the slope to differ from one random sample to another?
]

![](06-drawing-inference_files/figure-html/unnamed-chunk-11-1.png)

* the **standard error** of `\(\hat{\beta_1}\)` ( `\(SE_{\hat{\beta}_1}\)` ) is how much we expect the sample slope to vary from one random sample to another.

---

## Sparrows

.question[
How can we quantify how much we'd expect the slope to differ from one random sample to another?
]

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy()
```

```
## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    1.37     0.957       1.43 1.56e- 1
*## 2 WingLength     0.467    0.0347     13.5  2.62e-25
```
]

---

## Sparrows

.question[
We need a **test statistic** that incorporates `\(\hat{\beta}_1\)` and the standard error `\(SE_{\hat\beta_1}\)`
]

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy()
```

---

## Sparrows

.question[
We need a **test statistic** that incorporates `\(\hat{\beta}_1\)` and the standard error `\(SE_{\hat\beta_1}\)`
]

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy()
```

# `\(t = \frac{\hat\beta_1}{SE_{\hat\beta_1}}\)`

---

## Sparrows

.question[
We need a **test statistic** that incorporates `\(\hat{\beta}_1\)` and the standard error `\(SE_{\hat\beta_1}\)`
]

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy()
```

```r
0.467 / 0.0347
```

```
## [1] 13.45821
```

---

## Sparrows

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy()
```

```r
0.467 / 0.0347
```

```
## [1] 13.45821
```

---

## Sparrows

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy()
```

* "the sample slope is more than 13 standard errors above a slope of zero"

---

## Sparrows

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy()
```

---

## Sparrows

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy()
```

* confidence intervals
* p-values

---

## Sparrows

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
* tidy(conf.int = TRUE)
```

```
## # A tibble: 2 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)    1.37     0.957       1.43 1.56e- 1   -0.531     3.26 
*## 2 WingLength     0.467    0.0347     13.5  2.62e-25    0.399     0.536
```
]

* confidence intervals
* p-values

---

## Sparrows

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy(conf.int = TRUE)
```

* confidence intervals
* p-values

---

## Sparrows

.question[
What if we knew what the distribution of the "statistic" would be under the null hypothesis?
]

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy()
```

---

## Sparrows

```r
null_sparrow_data <- data.frame(
  WingLength = rnorm(10, 27, 4),
  Weight = rnorm(10, 14, 3)
)
lm(Weight ~ WingLength, data = null_sparrow_data) %>%
  tidy()
```

```
## # A tibble: 2 x 5
##   term        estimate std.error statistic p.value
##   <chr>          <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)    6.17      3.55       1.74  0.121 
## 2 WingLength     0.334     0.129      2.58  0.0326
```

---

## Sparrows

```r
null_sparrow_data <- data.frame(
* WingLength = rnorm(10, 27, 4),
  Weight = rnorm(10, 14, 3)
)
lm(Weight ~ WingLength, data = null_sparrow_data) %>%
  tidy()
```

```
## # A tibble: 2 x 5
##   term        estimate std.error statistic p.value
##   <chr>          <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)  11.2        5.96      1.88   0.0968
## 2 WingLength    0.0840     0.206     0.407  0.695
```

---

## Sparrows

```r
null_sparrow_data <- data.frame(
  WingLength = rnorm(10, 27, 4),
* Weight = rnorm(10, 14, 3)
)
lm(Weight ~ WingLength, data = null_sparrow_data) %>%
  tidy()
```

```
## # A tibble: 2 x 5
##   term        estimate std.error statistic p.value
##   <chr>          <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)   11.7       5.94      1.97   0.0846
## 2 WingLength     0.121     0.223     0.543  0.602
```

---

## Sparrows

```r
*gen_null_stat <- function() {
  null_sparrow_data <- data.frame(
    WingLength = rnorm(10, 27, 4),
    Weight = rnorm(10, 14, 3)
  )
  lm(Weight ~ WingLength, data = null_sparrow_data) %>%
    tidy() %>%
    filter(term == "WingLength") %>%
    select("statistic")
*} 
```
]

```r
gen_null_stat()
```

```
## # A tibble: 1 x 1
##   statistic
##       <dbl>
## 1    -0.661
```

---

## Sparrows

```r
gen_null_stat()
```

```
## # A tibble: 1 x 1
##   statistic
##       <dbl>
## 1    -0.422
```

```r
gen_null_stat()
```

```
## # A tibble: 1 x 1
##   statistic
##       <dbl>
## 1     0.536
```

```r
gen_null_stat()
```

```
## # A tibble: 1 x 1
##   statistic
##       <dbl>
## 1     -2.19
```
]

---

## Sparrows

```r
null_stats <- map_df(1:10000, ~ gen_null_stat())
```

---

## Sparrows

```r
null_stats <- map_df(1:10000, ~ gen_null_stat())
```

![](06-drawing-inference_files/figure-html/unnamed-chunk-33-1.png)

---

## Sparrows

![](06-drawing-inference_files/figure-html/unnamed-chunk-34-1.png)

---

## Sparrows

![](06-drawing-inference_files/figure-html/unnamed-chunk-35-1.png)

* Normal?
--

* What distribution is similar to the normal but with fatter tails?

---

## Sparrows

![](06-drawing-inference_files/figure-html/unnamed-chunk-36-1.png)

* the *t-distribution!*
--

* this is a **t-distribution** with **n-2** degrees of freedom.

---

## Sparrows

The distribution of test statistics we would expect given the **null hypothesis is true**, `\(\beta_1 = 0\)`, is **t-distribution** with **n-2** degrees of freedom.

![](06-drawing-inference_files/figure-html/unnamed-chunk-37-1.png)

---

## Sparrows

![](06-drawing-inference_files/figure-html/unnamed-chunk-38-1.png)

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy()
```

---

## Sparrows

![](06-drawing-inference_files/figure-html/unnamed-chunk-40-1.png)

---

## Sparrows

![](06-drawing-inference_files/figure-html/unnamed-chunk-41-1.png)

* p-value

---

# p-value

The probability of getting a statistic as extreme or more extreme than the observed test statistic **given the null hypothesis is true**

---

## Sparrows

![](06-drawing-inference_files/figure-html/unnamed-chunk-42-1.png)

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy()
```

---

## Return to generated data, n = 20

![](06-drawing-inference_files/figure-html/unnamed-chunk-45-1.png)

* Let's say we get a statistic of **1.5** in a sample
---

## Let's do it in R!

The proportion of area less than 1.5

![](06-drawing-inference_files/figure-html/unnamed-chunk-46-1.png)

```r
pt(1.5, df = 18)
```

```
## [1] 0.9245248
```

---

## Let's do it in R!

The proportion of area **greater** than 1.5

![](06-drawing-inference_files/figure-html/unnamed-chunk-48-1.png)

```r
pt(1.5, df = 18, lower.tail = FALSE)
```

```
## [1] 0.07547523
```
---

## Let's do it in R!

The proportion of area **greater** than 1.5 **or** **less** than -1.5.

![](06-drawing-inference_files/figure-html/unnamed-chunk-50-1.png)

```r
pt(1.5, df = 18, lower.tail = FALSE) * 2
```

```
## [1] 0.1509505
```

---

# p-value

The probability of getting a statistic as extreme or more extreme than the observed test statistic **given the null hypothesis is true**

---

## Hypothesis test

* **null hypothesis** `\(H_0: \beta_1 = 0\)` 
* **alternative hypothesis** `\(H_A: \beta_1 \ne 0\)`
--

* **p-value**: 0.15
--

* Often, we have an `\(\alpha\)`-level cutoff to compare this to, for example **0.05**. Since this is greater than **0.05**, we **fail to reject the null hypothesis**

---

# confidence intervals

If we use the same sampling method to select different samples and computed an interval estimate for each sample, we would expect the true population parameter ( `\(\beta_1\)` ) to fall within the interval estimates 95% of the time.

---

# Confidence interval

`\(\Huge \hat\beta_1 \pm t^∗ \times SE_{\hat\beta_1}\)`

]

* `\(t^*\)` is the critical value for the `\(t_{n−2}\)` density curve to obtain the desired confidence level
--

* Often we want a **95% confidence level**.

---

## Let's do it in R!

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy(conf.int = TRUE)
```

```r
qt(0.025, df = nrow(Sparrows) - 2, lower.tail = FALSE)
```

```
## [1] 1.980992
```
---

## Let's do it in R!

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy(conf.int = TRUE)
```

```r
qt(0.025, df = nrow(Sparrows) - 2, lower.tail = FALSE)
```

```
## [1] 1.980992
```

---

## Let's do it in R!

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy(conf.int = TRUE)
```

```r
qt(0.025, df = nrow(Sparrows) - 2, lower.tail = FALSE)
```

```
## [1] 1.980992
```

---

## Let's do it in R!

```r
lm(Weight ~ WingLength, data = Sparrows) %>%
  tidy(conf.int = TRUE)
```

```r
t_star <- qt(0.025, df = nrow(Sparrows) - 2, lower.tail = FALSE)
0.467 + t_star * 0.0347
```

```
## [1] 0.536
```

```r
0.467 - t_star * 0.0347
```

```
## [1] 0.398
```

---

# confidence intervals

---

## <i class="fas fa-laptop"></i> `Porsche Price (3)`

- Go to RStudio Cloud and open `Porsche Price (3)`