Logistic annotated example
Some background:
In simple linear regression the outcome variable Y is predicted from the equation:
in which b0 is the Y intercept, b1 quantifies the relationship between the predictor and outcome, X1 is
the value of the predictor variable and ε is an error term.
For several predictors, a similar model is used in which the outcome (Y) is predicted from a
combination of each predictor variable (X) multiplied by its respective regression coefficient (b) plus
ε the error term in the model.
Logistic regression is designed to use a mix of continuous and categorical predictor variables to
predict a binomial/dichotomous categorical dependent variable. For a binary response taking the
values 0 and 1 (e.g., absence of CHD and presence of CHD) the expected value is simply the
probability, p, that the variable takes the value one, i.e., the probability of having CHD.
We can’t model p directly as for linear regression two reasons
1. The observed values do not follow a normal distribution with mean p, but rather what is
known as a Bernoulli distribution
2. The probability of an event occurring must be between 0 and 1. Using a linear regression
model does not ensure that this is so.
It is more appropriate to model p indirectly via logit transformation of p i.e., ln[p/(1-p)]. Remember
that p/(1-p) is the odds of an event occurring, so we are in effect modelling the log-odds of an event
as a linear function of the explanatory variables. The parameters in a logistic regression model are
estimated by maximum likelihood. Loosely speaking, the likelihood of a set of data is the probability
of obtaining that particular set of data given the chosen probability model. This expression contains
the unknown parameters. Those values of the parameter that maximize the sample likelihood are
known as the maximum likelihood estimates. Using MLE, the parameters are chosen to maximise the
likelihood that the assumed model results in the observed data.
Instead, the logistic equation predicts the log-odds of the event of interest occurring. Specifically,
the general equation for logistic regression is:
Which is usually written as below.
The estimated regression coefficients in a logistic regression model give the estimated change in the
log-odds corresponding to a unit change in the corresponding explanatory variable conditional on
the other explanatory variables remaining constant. The parameters are usually exponentiated to
give results in terms of odds.
As an example, I have used a dataset consisting of the variables age (years), weight (kgs), gender
(0=female, male=1), VO2max and coronary heart disease status (0=no, 1=yes). I have attached this
dataset as an excel sheet “vo2max data.xlsx”. To perform logistic regression want to how well the
incidence of coronary heart disease can be predicted based of age, weight, gender and VO2max in a
sample of 100 persons.
These are the steps I would perform.
1. Exploratory data analysis.
This is to get a feel of your data e.g., what type are the variables (e.g., continuous,
dichotomous, ordinal etc, how are the values of each are distributed, missing values,
outliers).
Part 1: Descriptive Statistics for Continuous Variables
When summarizing a quantitative (continuous/interval/ratio) variable, we are typically interested in
things like:

How many observations were there? How many cases had missing values? (N valid; N
missing)

Where is the "center" of the data? (Mean, median)

Where are the "benchmarks" of the data? (Quartiles, percentiles)

How spread out is the data? (Standard deviation/variance)

What are the extremes of the data? (Minimum, maximum; Outliers)

What is the "shape" of the distribution? Is it symmetric or asymmetric? Are the values
mostly clustered about the mean, or are there many values in the "tails" of the distribution?
(Skewness, kurtosis)

Descriptives
Descriptives (Analyze > Descriptive Statistics > Descriptives) is best to obtain quick summaries of
numeric variables, or to compare several numeric variables side-by-side.
//
Then use Statistics tab and choose the following
You get this output:
Explore
Explore (Analyze > Descriptive Statistics > Explore) is best used to deeply investigate a single numeric
variable, with or without a categorical grouping variable. It can produce a large number of
descriptive statistics, as well as confidence intervals, normality tests, and plots.
To explore statistics of by CHD statistic.
EXAMINE VARIABLES=age weight gender vo2max BY chd
/PLOT BOXPLOT STEMLEAF HISTOGRAM NPPLOT
/COMPARE GROUPS
/STATISTICS DESCRIPTIVES EXTREME
/CINTERVAL 95
/MISSING LISTWISE
/NOTOTAL.
This is the output – some is more useful than other bits. I can go through this and explain and
interpret any output you’re unsure about and let you know the main things you need to be
looking for.
Explore
Notes
Output Created
02-MAR-2022 12:19:14
Comments
Input
Active Dataset
DataSet1
Filter
<none>
Weight
<none>
Split File
<none>
N of Rows in Working Data File
Missing Value Handling
Definition of Missing
98
User-defined missing values for
dependent variables are treated
as missing.
Cases Used
Statistics are based on cases
with no missing values for any
dependent variable or factor
used.
Syntax
EXAMINE VARIABLES=age
weight gender vo2max BY chd
/PLOT BOXPLOT STEMLEAF
HISTOGRAM NPPLOT
/COMPARE GROUPS
/STATISTICS
DESCRIPTIVES EXTREME
/CINTERVAL 95
/MISSING LISTWISE
/NOTOTAL.
Resources
Processor Time
00:00:06.14
Elapsed Time
00:00:04.30
chd
Case Processing Summary
Cases
Valid
chd
age
0
N
Missing
Percent
65
100.0%
N
Total
Percent
0
0.0%
N
Percent
65
100.0%
weight
gender
vo2max
1
33
100.0%
0
0.0%
33
100.0%
0
65
100.0%
0
0.0%
65
100.0%
1
33
100.0%
0
0.0%
33
100.0%
0
65
100.0%
0
0.0%
65
100.0%
1
33
100.0%
0
0.0%
33
100.0%
0
65
100.0%
0
0.0%
65
100.0%
1
33
100.0%
0
0.0%
33
100.0%
Descriptives
chd
age
0
Statistic
Mean
38.31
95% Confidence Interval for
Lower Bound
36.51
Mean
Upper Bound
40.11
5% Trimmed Mean
37.55
Median
36.00
Variance
7.271
Minimum
30
Maximum
63
Range
33
Interquartile Range
9
Skewness
1.532
.297
Kurtosis
2.643
.586
Mean
45.76
1.572
95% Confidence Interval for
Lower Bound
42.56
Mean
Upper Bound
48.96
5% Trimmed Mean
45.03
Median
43.00
Variance
81.564
Std. Deviation
weight
0
.902
52.873
Std. Deviation
1
Std. Error
9.031
Minimum
35
Maximum
74
Range
39
Interquartile Range
12
Skewness
1.256
.409
Kurtosis
1.611
.798
76.9705
1.70114
Mean
95% Confidence Interval for
Lower Bound
73.5720
Mean
Upper Bound
80.3689
5% Trimmed Mean
76.5597
Median
77.3700
Variance
188.102
Std. Deviation
1
13.71503
Minimum
50.00
Maximum
115.42
Range
65.42
Interquartile Range
17.02
Skewness
.401
.297
Kurtosis
.389
.586
86.3697
2.63141
Mean
95% Confidence Interval for
Lower Bound
81.0097
Mean
Upper Bound
91.7297
5% Trimmed Mean
86.6962
Median
88.8300
Variance
228.502
Std. Deviation
gender
0
15.11629
Minimum
53.43
Maximum
112.59
Range
59.16
Interquartile Range
22.18
Skewness
-.470
.409
Kurtosis
-.380
.798
.57
.062
Mean
95% Confidence Interval for
Lower Bound
.45
Mean
Upper Bound
.69
5% Trimmed Mean
.58
Median
1.00
Variance
.249
Std. Deviation
.499
Minimum
0
Maximum
1
Range
1
Interquartile Range
1
Skewness
Kurtosis
1
Mean
-.286
.297
-1.980
.586
.79
.072
95% Confidence Interval for
Lower Bound
.64
Mean
Upper Bound
.94
5% Trimmed Mean
Median
.82
1.00
Variance
.172
Std. Deviation
.415
Minimum
0
Maximum
1
Range
1
Interquartile Range
0
Skewness
Kurtosis
vo2max
0
Mean
.798
45.0000
1.14909
42.7044
Mean
Upper Bound
47.2956
5% Trimmed Mean
44.9076
Median
45.0600
85.827
9.26427
Minimum
27.35
Maximum
62.50
Range
35.15
Interquartile Range
13.17
Skewness
Kurtosis
Mean
.230
.297
-.801
.586
40.6242
1.03376
95% Confidence Interval for
Lower Bound
38.5185
Mean
Upper Bound
42.7299
5% Trimmed Mean
40.5282
Median
40.5800
Variance
35.266
Std. Deviation
5.93851
Minimum
28.30
Maximum
55.19
Range
26.89
Interquartile Range
8.95
Skewness
.191
.409
-.022
.798
Kurtosis
Extreme Values
chd
0
.187
Lower Bound
Std. Deviation
age
.409
95% Confidence Interval for
Variance
1
-1.476
Case Number
Highest
Value
1
74
63
2
85
63
Lowest
1
Highest
Lowest
weight
0
Highest
Lowest
1
Highest
Lowest
gender
0
Highest
3
65
55
4
76
51
5
35
50
1
67
30
2
63
31
3
57
31
4
43
31
5
41
31a
1
60
74
2
71
62
3
98
61
4
97
58
5
12
55b
1
9
35
2
32
36
3
25
36
4
4
36
5
45
37
1
48
115.42
2
15
111.80
3
5
103.00
4
83
101.62
5
43
97.75
1
58
50.00
2
49
51.96
3
88
53.00
4
67
55.00
5
35
58.07
1
59
112.59
2
94
111.98
3
90
103.53
4
64
103.23
5
40
101.25
1
25
53.43
2
17
56.18
3
32
62.59
4
55
64.00
5
62
68.29
1
1
1
2
2
1
Lowest
1
Highest
Lowest
vo2max
0
Highest
Lowest
1
Highest
Lowest
3
5
1
4
7
1
5
15
1c
1
91
0
2
88
0
3
86
0
4
78
0
5
74
0d
1
3
1
2
9
1
3
11
1
4
17
1
5
20
1c
1
87
0
2
62
0
3
60
0
4
55
0
5
31
0d
1
23
62.50
2
38
62.13
3
77
61.76
4
52
61.71
5
70
60.92
1
36
27.35
2
91
30.37
3
8
30.38
4
15
31.94
5
74
31.99
1
27
55.19
2
82
50.19
3
95
49.22
4
32
48.23
5
17
47.23
1
4
28.30
2
98
32.00
3
71
32.00
4
87
33.67
5
94
33.73
a. Only a partial list of cases with the value 31 are shown in the table of
lower extremes.
b. Only a partial list of cases with the value 55 are shown in the table of
upper extremes.
c. Only a partial list of cases with the value 1 are shown in the table of upper
extremes.
d. Only a partial list of cases with the value 0 are shown in the table of lower
extremes.
Tests of Normality
Kolmogorov-Smirnova
chd
age
weight
gender
vo2max
Statistic
df
Shapiro-Wilk
Sig.
Statistic
df
Sig.
0
.148
65
.001
.854
65
.000
1
.165
33
.022
.893
33
.003
0
.082
65
.200*
.981
65
.424
1
.200
33
.002
.952
33
.154
0
.375
65
.000
.630
65
.000
1
.483
33
.000
.505
33
.000
0
.067
65
.200*
.967
65
.082
1
.074
33
.200*
.991
33
.993
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction
age
Histograms
Stem-and-Leaf Plots
age Stem-and-Leaf Plot for
chd= 0
Frequency
26.00
16.00
13.00
5.00
Stem &
3
3
4
4
.
.
.
.
Leaf
01111112222222333333444444
5555666677789999
0000111233334
66888
2.00
5 .
3.00 Extremes
Stem width:
Each leaf:
01
(>=55)
10
1 case(s)
age Stem-and-Leaf Plot for
chd= 1
Frequency
Stem &
.00
3
8.00
3
10.00
4
6.00
4
2.00
5
4.00
5
2.00
6
1.00 Extremes
Stem width:
Each leaf:
.
.
.
.
.
.
.
Leaf
56667888
0011122233
566779
01
5558
12
(>=74)
10
Normal Q-Q Plots
1 case(s)
Detrended Normal Q-Q Plots
weight
Histograms
Stem-and-Leaf Plots
weight Stem-and-Leaf Plot for
chd= 0
Frequency
8.00
Stem &
5 .
Leaf
01358889
11.00
6
23.00
7
13.00
8
6.00
9
2.00
10
2.00 Extremes
Stem width:
Each leaf:
.
.
.
.
.
02223488899
00122234455567888899999
0112225567779
002457
13
(>=112)
10.00
1 case(s)
weight Stem-and-Leaf Plot for
chd= 1
Frequency
2.00
3.00
6.00
7.00
9.00
4.00
2.00
Stem width:
Each leaf:
Stem &
5
6
7
8
9
10
11
.
.
.
.
.
.
.
Leaf
36
248
233445
7888889
013444577
1133
12
10.00
1 case(s)
Normal Q-Q Plots
Detrended Normal Q-Q Plots
gender
Histograms
Stem-and-Leaf Plots
gender Stem-and-Leaf Plot for
chd= 0
Frequency
Stem &
28.00
.00
.00
.00
.00
37.00
Stem width:
Each leaf:
0
0
0
0
0
1
.
.
.
.
.
.
Leaf
0000000000000000000000000000
0000000000000000000000000000000000000
1
1 case(s)
gender Stem-and-Leaf Plot for
chd= 1
Frequency
Stem &
7.00 Extremes
26.00
1 .
Stem width:
Each leaf:
Leaf
(=<.0)
00000000000000000000000000
1
Normal Q-Q Plots
1 case(s)
Detrended Normal Q-Q Plots
vo2max
Histograms
Stem-and-Leaf Plots
vo2max Stem-and-Leaf Plot for
chd= 0
Frequency
1.00
10.00
9.00
12.00
16.00
5.00
5.00
7.00
Stem width:
Each leaf:
Stem &
2
3
3
4
4
5
5
6
.
.
.
.
.
.
.
.
Leaf
7
0011333344
666667788
000011222344
5555577778899999
00124
55578
0001122
10.00
1 case(s)
vo2max Stem-and-Leaf Plot for
chd= 1
Frequency
1.00
4.00
9.00
12.00
5.00
1.00
1.00
Stem width:
Each leaf:
Stem &
2
3
3
4
4
5
5
.
.
.
.
.
.
.
Leaf
8
2233
555577889
000012222444
55789
0
5
10.00
Normal Q-Q Plots
1 case(s)
Detrended Normal Q-Q Plots

Frequencies Part I (Continuous Variables)
Frequencies (Analyze > Descriptive Statistics > Frequencies) is typically used to analyze categorical
variables, but can also be used to obtain percentile statistics that aren't otherwise included in the
Descriptives, Compare Means, or Explore procedures.
Part 2: Descriptive Statistics for Categorical Variables
When summarizing qualitative (nominal or ordinal) variables, we are typically interested in things
like:

How many cases were in each category? (Counts)

What proportion of the cases were in each category? (Percentage, valid percent, cumulative
percent)

What was the most frequently occurring category (i.e., the category with the most
observations)? (Mode)
In Part 2, we describe how to obtain descriptive statistics for categorical variables using the
Frequencies and Crosstabs procedures.

Frequencies Part II (Categorical Variables)
Frequencies (Analyze > Descriptive Statistics > Frequencies) is primarily used to create frequency
tables, bar charts, and pie charts for a single categorical variable.

Crosstabs
The Crosstabs procedure (Analyze > Descriptive Statistics > Crosstabs) is used to create contingency
tables, which describe the interaction between two categorical variables. This tutorial covers the
descriptive statistics aspects of the Crosstabs procedure, including and row, column, and total
percents.

Multiple Response Sets / Working with "Check All That Apply" Survey Data
Check-all-that-apply questions on surveys are recorded as a set of binary indicator variables for each
checkbox option. Frequency tables and crosstabs alone don't capture the dependent nature of this
data -- and that's where Multiple Response Sets come in.