41.0
42.0
43.0
44.0
45.0
46.0
47.0
48.0
49.0
50.0
51.0
52.0
53.0
54.0
55.0
56.0
57.0
58.0
60.0
62.0
29.0
30.0
31.0
32.0
33.0
34.0
35.0
36.0
37.0
38.0
39.0
40.0
18.0
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
Dependent Variable: Outcome of Field Goal Attempt (1=Success,
0=Failure)
Independent Variable: Distance (Yards)
Data:
YARDS * SUCCESS Crosstabulation
Count
YARDS
Total
36
29
25
18
31
21
34
27
14
14
8
6
1
948
1
2
1
2
27
25
32
33
26
18
33
25
33
29
32
28
28
Total
1
33
30
17
27
5
19
18
36
32
32
37
22
SUCCESS
0
15
13
11
6
11
8
9
7
5
5
9
7
1
192
0
2
0
1
6
7
7
4
3
6
3
2
10
2
7
5
7
0
0
1
0
2
1
0
0
1
4
3
0
1
1
21
16
14
12
20
13
25
20
3
1
5
7
0
756
1
0
1
1
20
21
26
26
23
16
30
19
26
24
22
26
21
1
33
28
17
26
5
18
17
36
32
31
33
19

Random Component: Binomial Distribution
Systematic Component: g(

) =

+

X
Link Function: logit g(

) = log



1
 


Regression Equation:

( x )

1 e
   x
 e
   x
Estimated Regression Equation:
Variables in the Equation
1
YARDS
Constant
B
-.110
5.698
S.E.
.011
.451
a. Variable(s) entered on s tep 1: YARDS.
Wald
107.856
159.545
df
1
1
Sig.
.000
.000
Exp(B)
.896
298.235
95.0% C.I.for EXP(B)
Lower
.878
Upper
.915

^
( x )
1 e

5 .
698

0 .
110 x
 e
5 .
698

0 .
110 x
Yardage Probability
20
25
30
35
40
45
50
55
.9706
.9502
.9167
.8639
.7855
.6787
.5493
.4129
95% CI for

: -0.110

1.96(0.011)

-0.110

0.022

(-0.132 , -0.088)

Odds Ratio: OR
 odds ( x

1 ) odds ( x )



1

( x
 
(
 x
1 )

1 )




1


(
 x )
( x )


 e

Estimated odds ratio: e
^
  e

0 .
110 
0 .
896
Odds of successful fieldgoal change by about (0.896-1)100% = -10.4% for each extra yard of the attempt.
95% CI for odds ratio: (e -0.132
, e -0.088
)

(0.876 , 0.916)
Hosmer-Lemeshow Goodness-of-Fit Test:
1.
Lump attempts into 10 bins of approximately equal number of attempts
2.
Obtain the observed and predicted (expected) numbers of successes and failures for each bin (20 total cells)
3.
For each bin, compute the contribution to the chi-square statistic:
X
2 
( observed expected)
2 expected
4.
Sum the results from step 3 over all 20 cells
Hosme r and Leme show Test
St ep
1
Chi-square
5.974
df
8
Sig.
.650
Contingency Table for Hosmer and Lemeshow Test
SUCCESS = 0 SUCCESS = 1
Step
1
1
2
Observed Expected
47 47.043
35 36.501
Observed Expected
45 44.957
57 55.499
Total
92
92
3 28 27.657
58 58.343
86
4 19 22.232
69 65.768
88
5 19 18.588
76 76.412
95
6 17 12.632
67 71.368
84
7 14 10.995
88 91.005
102
8 8 6.779
83 84.221
91
9 3 5.638
103 100.362
106
10 2 3.936
110 108.064
112
Don’t reject the null hypothesis that the model fit is appropriate.

Computational Approach to Obtaining Logistic Regression Analysis
Data: n i
observations at the i th of m distinct levels of the independent variable(s), with y i
successes.
X






 x x

'
1
2
' x m
'





 x i
' 

1 x
1 i
 x pi

 










0
1 p






Note: p =1 in this case
With Likelihood and log-Likelihood Functions:
L (

| ( n
1
, y
1
),..., ( n m
, y m
)) log( L )
 i n 

1
 log( n i
!
)
 log(
 i m 

1

 y i
!
)
 y i
!
( n i n i
!
 log(( n i
 y i
)!




1 exp(
 x i
' exp(
 x i
'
)

)

 y i y i
)!
)

 i n 

1 y i x i
'
 


1
 n  i

1 n i
1 exp( x i
' 
)

 n i
 y i log

1
 exp( x i
'

)

The derivative of the log-likelihood wrt

:
 log( L )
 
  y i x i
  n i
1

1 exp( x i
'

) x i exp( x i
' 
)
 
( y i
 n i
 i
) x i
 g (

)
The Hessian matrix:
 2
'
   n x i

 x i
'

)
 x i
' exp( x i
'
  x i
'
 x i x i
'

)

 x i
'

)

2
   n x x i i i
' exp( x i
'

) 
W ii
 n
 i i
(1
  i
) W ij
1
 x i
'

)

2

0 i
 j
 
Newton-Raphson-Algorithm:

^
NEW

^

OLD


 G


^

OLD 




1 g



^
OLD 

'

G

Results from SAS PROC IML:
BETA_NEW SEBETA
5.6978802 0.4510991
-0.10991 0.0105832
VBETA LOGLIKE
0.2034904 -0.004683 -408.6017
-0.004683 0.000112
Note: There log-likelihood function evaluation does not match SPSS’ (it does not evaluate the first term, which does not involve

). Any tests concerning

will not be effected.
1 y
.8
.6
.4
.2
20 30 40 50 60 70 x
For more detail, see Agresti (2002), Categorical Data Analysis . Chapter 4.
Sources: www.jt-sw.com
, www.espn.com