Transcript A Broad Overview of Key Statistical Concepts

Inverse prediction
Using a regression model to predict
the X value that produced a new Y
value…
Examples
• Predict the actual steps taken (X) that
produced the pedometer reading (Y).
• Predict the drug dose (X) that produced the
decrease in cholesterol level (Y).
• Predict industrial plant’s steam usage (X)
that produced an average atmospheric
temperature (Y).
To predict Xh(new), just solve for it…
Yˆh( new )  b0  b1 X
Algebra
Xˆ h( new ) 
Yh( new )  b0
b1
… providing b1 is not equal to 0.
(Approximate) confidence limits
 

ˆ
X h( new )  t 1  ; n  2   s predX
 2

where:
s( predX) 

MSE  1  s b1  Xˆ h( new )  X
1   
2
b1  n 
b12
2

2
Example: Laura’s step and
pedometer data
steps
60
95
95
85
70
50
55
80
75
65
85
80
50
75
ped
61
94
92
85
69
49
54
77
75
64
85
76
49
74
random
0.026
0.166
0.213
0.269
0.322
0.430
0.481
0.548
0.613
0.678
0.742
0.800
0.866
0.911
• steps = number of actual steps
taken (determined by counting)
• ped = number of steps reported by
Freestyle hip pedometer
• random = uniform random
numbers between 0 and 1, sorted to
dictate order in which data collected
Fitted line plot
Regression Plot
pedometer = 0.791866 + 0.973445 steps
S = 1.34192
R-Sq = 99.2 %
R-Sq(adj) = 99.2 %
Pedometer steps
90
80
70
60
50
50
60
70
?
80
Actual steps
90
How many steps have I actually
taken when the pedometer tells me
I’ve taken 80 steps?
Ped  0.79  0.97 steps
Ped  0.79 80  0.79
Steps 

 81.6
0.97
0.97
I’m getting robbed 1.6 steps for every 80 steps I take!
Mean of steps = 72.857
Predictor
Constant
steps
Analysis of
Source
Regression
Error
Total
Coef
0.792
0.97344
SE Coef
1.825
0.02456
Variance
DF
SS
1
2829.2
12
21.6
13
2850.9
MS
2829.2
1.8
T
0.43
39.64
F
1571.16
P
0.672
0.000
P
0.000
1.8 
1  0.0246  80  72.86 
s predX  
1  
 1.44
2 
2
0.973  14 
0.973
2
2
I want to be 95% confident…
Xˆ h( new )  t 0.975;12 s predX
81.6  2.1791.44
81.6  3.14
I am 95% confident that I actually took between 78.5 and
84.7 steps when my pedometer says that I took 80 steps.