-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathweek12-part2.html
More file actions
414 lines (326 loc) · 13.5 KB
/
week12-part2.html
File metadata and controls
414 lines (326 loc) · 13.5 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>Defining and Detecting Multicollinearity</title>
<meta charset="utf-8" />
<meta name="author" content="STAT 021 with Prof Suzy" />
<script src="week12-part2_files/header-attrs-2.14/header-attrs.js"></script>
<link href="week12-part2_files/remark-css-0.0.1/default.css" rel="stylesheet" />
<link href="week12-part2_files/remark-css-0.0.1/default-fonts.css" rel="stylesheet" />
</head>
<body>
<textarea id="source">
class: center, middle, inverse, title-slide
# Defining and Detecting Multicollinearity
### STAT 021 with Prof Suzy
### Swarthmore College
---
<style type="text/css">
pre {
background: #FFBB33;
max-width: 100%;
overflow-x: scroll;
}
.scroll-output {
height: 70%;
overflow-y: scroll;
}
.scroll-small {
height: 50%;
overflow-y: scroll;
}
.red{color: #ce151e;}
.green{color: #26b421;}
.blue{color: #426EF0;}
</style>
## Multicollinearity
There are a lot of important c-words in statistics. Especially confusing can be the following: collinearity, correlation, and covariance. But, somewhat intuitively, there is a relationship among these three terms.
Recall the definitions of correlation and covariance for any two random variables, say, `\(X\)` and `\(Y\)`:
`$$Cov(X,Y) = E[(X -E[X])(Y-E[Y])] = \dots = E[XY] - E[X]E[Y]$$`
`$$Cor(X,Y) = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}$$`
--
Also, recall the fact that .blue[if `\(X\)` and `\(Y\)` are independent, then `\(Cov(X,Y)=0\)` and therefore `\(X\)` and `\(Y\)` are uncorrelated.]
**But** if `\(X\)` and `\(Y\)` are uncorrelated, then it is still possible for `\(X\)` and `\(Y\)` to be dependent or independent.
---
## Multicollinearity
So now we see that correlation is just a standardized version of the covariance between two variables. (Standardized in the sense that it will always be between the interval `\([-1,1]\)`.)
**Q:** What is multicollinearity?
--
This is a term specific to MLR models that describes the statistical phenomenon in which two or more predictor variables are highly correlated with each other.
In other words, this means that two predictor variables, say, `\(x_1\)` and `\(x_2\)`, are collinear if `\(Cor(x_1, x_2) \approx \pm 1\)`.
- This means that we could predict the values of one from the other in a SLR model!
- This is **not** a problem for .blue[estimation] or .blue[prediction] with a MLR.
- This **is** a problem however that inflates the estimated variances of our regression coefficients and therefore the individual test of significant slope parameters. Thus it can be a problem for .red[inference].<sup>[5]</sup>
- Outliers can have a big impact on the collinearity of a pair of variables.<sup>[5]</sup>
---
## Multicollinearity Example
### Public School SAT data
.scroll-output[Again let's investigate the MLR model with four predictor variables that we built to predict SAT scores of public schools. Does it seem like any of the predictor variables might be correlated with each other?
```r
SAT_data <- read_table2("Data/sat_data.txt", col_names=FALSE, cols(col_character(), col_double(), col_double(), col_double(), col_double(), col_double(), col_double(), col_double()))
colnames(SAT_data) = c("State", "PerPupilSpending", "StuTeachRatio", "Salary", "PropnStu", "SAT_verbal", "SAT_math", "SAT_tot")
SAT_data <- SAT_data %>% mutate(prop_taking_SAT = PropnStu/100) %>% select(-PropnStu)
SAT_data %>% select(c(PerPupilSpending, StuTeachRatio, Salary, prop_taking_SAT)) %>% plot
```
<!-- -->
]
---
## Multicollinearity Example
### Public School SAT data
.scroll-output[Again let's investigate the MLR model with four predictor variables that we built to predict SAT scores of public schools. Does it seem like any of the predictor variables might be correlated with each other?
```r
MLR_SAT4 <- lm(SAT_tot ~ PerPupilSpending +
StuTeachRatio +
Salary +
prop_taking_SAT, data = SAT_data)
summary(MLR_SAT4)
```
```
##
## Call:
## lm(formula = SAT_tot ~ PerPupilSpending + StuTeachRatio + Salary +
## prop_taking_SAT, data = SAT_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -90.531 -20.855 -1.746 15.979 66.571
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1045.972 52.870 19.784 < 2e-16 ***
## PerPupilSpending 4.463 10.547 0.423 0.674
## StuTeachRatio -3.624 3.215 -1.127 0.266
## Salary 1.638 2.387 0.686 0.496
## prop_taking_SAT -290.448 23.126 -12.559 2.61e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 32.7 on 45 degrees of freedom
## Multiple R-squared: 0.8246, Adjusted R-squared: 0.809
## F-statistic: 52.88 on 4 and 45 DF, p-value: < 2.2e-16
```
]
---
## Multicollinearity Example
### Public School SAT data
We can estimate the correlations among each pair of variables (without risking multiple testing issues):
.scroll-small[
```r
cor(SAT_data$PerPupilSpending, SAT_data$StuTeachRatio)
```
```
## [1] -0.3710254
```
```r
cor(SAT_data$PerPupilSpending, SAT_data$Salary)
```
```
## [1] 0.8698015
```
```r
cor(SAT_data$PerPupilSpending, SAT_data$prop_taking_SAT)
```
```
## [1] 0.5926274
```
```r
cor(SAT_data$StuTeachRatio, SAT_data$Salary)
```
```
## [1] -0.001146081
```
```r
cor(SAT_data$StuTeachRatio, SAT_data$prop_taking_SAT)
```
```
## [1] -0.2130536
```
```r
cor(SAT_data$Salary, SAT_data$prop_taking_SAT)
```
```
## [1] 0.6167799
```
Note: The estimated correlation between two variables is not affected by whether or not those variables are standardized because correlation itself is already standardized.
]
---
## Multicollinearity
When we see evidence of collinear predictor variables, it's a good idea to re-visit which variables you want to include in the model .red[because] collinearity affects the variance of our predictors and therefore affects the conclusion of the individual t-tests!<sup>[4]</sup>
**Q:** Do we need to be concerned about multiple testing with interaction terms? with collinearity?
--
**A:** We do need to keep an eye on multiple tests issues when we're determining which main effects and interaction effects to include in the model (these are questions of statistical significance). The problem of multiple testing is not a concern however, in estimation/prediction problems therefore it is not a major concern when checking for collinearity (all we're doing is getting estimates for different correlations).
**Note:** Interaction terms are structurally collinear, which we can't do much about.
---
## Multicollinearity
### Remedies
A couple of simple remedies for collinearity are
- Try collecting more data to see if the collinearity is dues to insufficient data;
- Reduce the number of variables in your model so that none of them are collinear;
- Other regression methods such as *ridge regression* (you don't need to know what this is for my class).<sup>[5]</sup>
---
## Multicollinearity
### Public School SAT data
Finally we can get a better model for the SAT data by adjusting which variables we include:
.scroll-small[
```
##
## Call:
## lm(formula = SAT_tot ~ PerPupilSpending + StuTeachRatio + prop_taking_SAT,
## data = SAT_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -92.284 -21.130 1.414 16.709 66.073
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1035.474 50.316 20.580 <2e-16 ***
## PerPupilSpending 11.014 4.452 2.474 0.0171 *
## StuTeachRatio -2.028 2.207 -0.919 0.3629
## prop_taking_SAT -284.912 21.548 -13.222 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 32.51 on 46 degrees of freedom
## Multiple R-squared: 0.8227, Adjusted R-squared: 0.8112
## F-statistic: 71.16 on 3 and 46 DF, p-value: < 2.2e-16
```
]
---
## Interaction terms and multicollinearity
Broadly speaking, there are two main types of collinearity:
1. Structural multicollinearity which occurs when we create a model term using other terms in the model (e.g. including interaction terms). This is a (unfortunately unavoidable) byproduct of the model that we specify.
2. Data multicollinearity which occurs when the data itself for different predictor variables are highly correlated. Some ways we can address this data-inherent multicollinearity is to collect more data or see if it makes sense to drop a variable from the model.
.footnote[Source: https://statisticsbyjim.com/regression/multicollinearity-in-regression-analysis/]
---
## Multicollinearity
### Some things to think about
- What are the effects of incorporating interaction terms in your linear model?
- What are the effects of severe multicollinearity among some predictor variables in your model?
***
### Reading along in your textbook
Chapter 3 Sections 10 and 11 and Chapter 9
</textarea>
<style data-target="print-only">@media screen {.remark-slide-container{display:block;}.remark-slide-scaler{box-shadow:none;}}</style>
<script src="https://remarkjs.com/downloads/remark-latest.min.js"></script>
<script>var slideshow = remark.create({
"highlightStyle": "github",
"highlightLines": true,
"countIncrementalSlides": false,
"ratio": "16:9",
"navigation": {
"scroll": false
}
});
if (window.HTMLWidgets) slideshow.on('afterShowSlide', function (slide) {
window.dispatchEvent(new Event('resize'));
});
(function(d) {
var s = d.createElement("style"), r = d.querySelector(".remark-slide-scaler");
if (!r) return;
s.type = "text/css"; s.innerHTML = "@page {size: " + r.style.width + " " + r.style.height +"; }";
d.head.appendChild(s);
})(document);
(function(d) {
var el = d.getElementsByClassName("remark-slides-area");
if (!el) return;
var slide, slides = slideshow.getSlides(), els = el[0].children;
for (var i = 1; i < slides.length; i++) {
slide = slides[i];
if (slide.properties.continued === "true" || slide.properties.count === "false") {
els[i - 1].className += ' has-continuation';
}
}
var s = d.createElement("style");
s.type = "text/css"; s.innerHTML = "@media print { .has-continuation { display: none; } }";
d.head.appendChild(s);
})(document);
// delete the temporary CSS (for displaying all slides initially) when the user
// starts to view slides
(function() {
var deleted = false;
slideshow.on('beforeShowSlide', function(slide) {
if (deleted) return;
var sheets = document.styleSheets, node;
for (var i = 0; i < sheets.length; i++) {
node = sheets[i].ownerNode;
if (node.dataset["target"] !== "print-only") continue;
node.parentNode.removeChild(node);
}
deleted = true;
});
})();
(function() {
"use strict"
// Replace <script> tags in slides area to make them executable
var scripts = document.querySelectorAll(
'.remark-slides-area .remark-slide-container script'
);
if (!scripts.length) return;
for (var i = 0; i < scripts.length; i++) {
var s = document.createElement('script');
var code = document.createTextNode(scripts[i].textContent);
s.appendChild(code);
var scriptAttrs = scripts[i].attributes;
for (var j = 0; j < scriptAttrs.length; j++) {
s.setAttribute(scriptAttrs[j].name, scriptAttrs[j].value);
}
scripts[i].parentElement.replaceChild(s, scripts[i]);
}
})();
(function() {
var links = document.getElementsByTagName('a');
for (var i = 0; i < links.length; i++) {
if (/^(https?:)?\/\//.test(links[i].getAttribute('href'))) {
links[i].target = '_blank';
}
}
})();
// adds .remark-code-has-line-highlighted class to <pre> parent elements
// of code chunks containing highlighted lines with class .remark-code-line-highlighted
(function(d) {
const hlines = d.querySelectorAll('.remark-code-line-highlighted');
const preParents = [];
const findPreParent = function(line, p = 0) {
if (p > 1) return null; // traverse up no further than grandparent
const el = line.parentElement;
return el.tagName === "PRE" ? el : findPreParent(el, ++p);
};
for (let line of hlines) {
let pre = findPreParent(line);
if (pre && !preParents.includes(pre)) preParents.push(pre);
}
preParents.forEach(p => p.classList.add("remark-code-has-line-highlighted"));
})(document);</script>
<script>
slideshow._releaseMath = function(el) {
var i, text, code, codes = el.getElementsByTagName('code');
for (i = 0; i < codes.length;) {
code = codes[i];
if (code.parentNode.tagName !== 'PRE' && code.childElementCount === 0) {
text = code.textContent;
if (/^\\\((.|\s)+\\\)$/.test(text) || /^\\\[(.|\s)+\\\]$/.test(text) ||
/^\$\$(.|\s)+\$\$$/.test(text) ||
/^\\begin\{([^}]+)\}(.|\s)+\\end\{[^}]+\}$/.test(text)) {
code.outerHTML = code.innerHTML; // remove <code></code>
continue;
}
}
i++;
}
};
slideshow._releaseMath(document);
</script>
<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
(function () {
var script = document.createElement('script');
script.type = 'text/javascript';
script.src = 'https://mathjax.rstudio.com/latest/MathJax.js?config=TeX-MML-AM_CHTML';
if (location.protocol !== 'file:' && /^https?:/.test(script.src))
script.src = script.src.replace(/^https?:/, '');
document.getElementsByTagName('head')[0].appendChild(script);
})();
</script>
</body>
</html>