Intermediate Methods in Epidemiology
2008
Exercise No. 2 - Measures of Disease Frequency and Association
Main topics covered in this laboratory exercise:
I.
The relationship of incidence rates and ratios to prevalence rates and ratios and how
these relate to the interpretation of cross-sectional studies.
II.
Calculation of
a. Relative Risk and Relative Odds: Similarities and differences
b. Relative odds in matched and non-matched samples
c. Confidence limits for relative odds and relative risks
III.
Case-Control Studies
a. Assumptions
b. Controls
1. Non-cases
2. Population sample
IV.
Random variability vs. bias
V.
Matched studies
a. Advantages and disadvantages
b. Assumptions
VI.
When and how to use a statistical significance trend test.
Department of Epidemiology - Johns Hopkins University - Copyright 1999
This laboratory exercise assumes familiarity with the following concepts:
Incidence and prevalence
Case fatality rate
Relative risk and relative odds
Cohort and case-control study design
Matched case-control design
Confounding
Matching as a method of confounder control
Sampling variability
How to label graphs
2
PART I: MEASURES OF DISEASE FREQUENCY
1.
BACKGROUND INFORMATION
The material for this part of the exercise will all be taken from a long-term study of
tuberculosis in Muscogee County, Georgia. In 1946, the county had a population of about
100,000 consisting of a city of 75,000 population and surrounding suburban and rural areas.
The city is industrial in character and is adjacent to a large military post. Thirty per cent of the
population was black.
Survey and Follow-up Procedures
In May and June, 1946, a combined tuberculosis-venereal disease survey aimed
primarily at persons over 15 years of age was conducted by the local and state health
departments with federal aid. Three months later a special census of the county population was
made which permitted a better estimation of the completeness of coverage than the Federal
Census figures because individuals enumerated in the special census could be matched against
those reached in the survey, and because the 1940 Federal Census was rather badly out of
date.
A 70 mm photofluorograph1 was offered to each participant over the age of 12 years.
Each photofluorograph was interpreted independently by two readers and all suspects of either
reader were requested to return for a standard 14 x 17 inch chest radiograph.
Persons classified as tuberculosis cases or suspects after the 14 x 17 inch X-ray were
advised to remain under observation for at least five years. During this period examinations
were to be repeated at least every three months. Only a few persons with clear-cut evidence
that the suspected abnormality was not tuberculous were discharged from follow-up before the
end of the five year period.
Tuberculin tests were done on most cases and suspects, and skin tests with fungal
antigens were done where indicated. Sputum examinations, including cultures, were done on
about one-half of the cases and suspects, including nearly all who were producing sputum or
were suspected of having active disease. Gastric washings were not obtained.
1
A photofluorograph (PF) is a photograph of an image on a fluorescent screen produced by X-rays.
Photofluorography of the chest is highly automated and rapid, and when used on a mass scale, is usually
inexpensive. A chest photofluorogram has been shown to reveal chest abnormalities as accurately as a
full-sized chest radiograph (Birkelo et al., JAMA, 133:359, 1947). Usually however, an abnormality
detected by photofluorography is checked by a full-sized chest radiograph.
3
On the average, each patient had more than 12 clinic visits, with radiographs, up to July
1, 1952. Sixty per cent of the persons initially classified as having tuberculous were examined in
the 5th year after the survey; 11 per cent had died; 8 per cent had moved away; and 21 per cent
were not examined in that year. Almost all of the last group were known to be living and
apparently well.
Case-finding Subsequent to the Survey
In 1950, 75,000 persons were radiographed in a second mass survey of the metropolitan
area exclusive of the military post. In addition, approximately 22,000 screening films were made
each year as a result of the pre-employment and food-handler requirements, prenatal services,
referral by private physicians, or other resources. Death certificates, hospital records and other
sources of medical information were continuously scanned for information on known cases and
suspects, as well as for possible unreported or unrecognized cases. Reporting is quite complete
in this community, and it can be assumed that all diagnosed tuberculosis came to the attention
of the health department, except for persons who had moved away prior to their diagnosis.
During the five-year period following the 1946 survey, new cases of tuberculosis were
matched to the file of 1946 survey participants. Observations on these cases were continued to
June 1952 to give even the most recently diagnosed case a follow-up period of at least one year.
Tuberculosis deaths among the cases diagnosed in the 1946 survey were also counted to June
1952.
Migration from the Community
To estimate the number of persons remaining in the area in 1952, a sample,
systematically selected, was visited in 1961 as part of a study of blood pressure levels. The
proportion of persons who had left the county is shown in Table 1. The very few individuals
whose whereabouts could not be ascertained were counted as having moved.
( 1) From the data in table 1, briefly describe the differences between individuals
who emigrate and those who do not in this population. What kind of
implications could these differences have for epidemiologic studies based on
this population?
In addition, the residence status of a systematic sample of participants in the 1950
survey who were then over 20 years of age was investigated in 1964. Sixty-six percent were
found to be still residing in the study area, with no differences between those who reacted to
tuberculin in 1950 and those who did not. (A reactor in this instance was defined as a person
having five or more mm of induration to 5 TU of tuberculin.)
4
Table 1
Emigration between Sept. 1, 1946 and July 1, 1961 in a sample of the surveyed population, by
race, sex, age and subcutaneous fatness in 1946, Muscogee County, Georgia.
Number
in sample
Per cent
emigrated
464
30
95
35
183
36
67
24
119
27
15-34
262
39
35-54
168
23
55+
34
15
0-4 mm.
157
35
5-9 mm.
216
32
10+ mm.
91
22
Characteristics
Total
Race-Sex
White Males
White Females
Black Males
Black Females
Age in 1946
Fat thickness in mm.
over trapezius muscle*
* Percentages adjusted to race-sex-age composition of total examined
population.
5
PREVALENCE vs. INCIDENCE
Strictly speaking the term relative risk is defined as the ratio of two incidence risks.
However, this term is commonly used with many quantities which approximate this ratio such as
the ratio of two rates, odds, or prevalences. This laboratory exercise uses the term "relative risk"
in this looser sense, however it is important to note how each estimate of the relative risk is
calculated. More specific uses of these and related terms are described in the relevant lecture.
Prevalence of Tuberculosis
Table 2 presents the number of persons screened, by race and age, and the cases of
pulmonary tuberculosis determined at the end of the 5-year observation period to have had
tuberculosis when surveyed. The tuberculosis deaths among these cases to June 1952 is also
given.
Table 2
Prevalence of pulmonary tuberculosis in 1946 survey, by race and age, Muscogee County,
Georgia.
White
Black
Total
<45
45+
<45
45+
Population screened
in survey
38,190
17,699
4,939
12,336
3,216
Cases of tuberculosis
568
134
245
119
70
Prevalence/1000
14.9
7.6
49.6
21.8
6.53
2.87
Prevalence Ratio
1.0
Tuberculosis deaths to
June 1952
34
2
Case fatality, %
6.0
1.49
Incidence of New Cases
6
5
2.04
16
11
15.71
In similar fashion, the number of new cases known to have developed among the
screened population during the next five years, together with the tuberculosis deaths among
them and the mid-point populations at risk are shown in Table 3.
Table 3
Incidence of new cases of pulmonary tuberculosis per 1000 estimated survey-negative
population during a 5-year period, 1946-1951, Muscogee County, Georgia.
White
Black
Total
<45
45+
<45
45+
33,656
15,554
4,348
10,915
2,839
New Cases of
tuberculosis
110
22
66
10
Incidence/1000/year
0.65
.28
Estimated midpoint
population
12
.70
.55
Relative Risk
1.0
Tuberculosis deaths
to June 1952
Case fatality, %
31
28.2
1.96
3
1
13.6
8.3
2.50
22
5
50.0
( 2) Calculate the prevalence, incidence and case fatality for younger blacks in
Tables 2 and 3. Note that for Table 3, the new cases of tuberculosis were
accumulated over a five year period so that the calculation of the annual
incidence rate must account for this. Do you observe the same patterns of
associations of race and age with tuberculosis prevalence and case fatality
from Table 2 (cross-sectional data) as you do with tuberculosis incidence
and case fatality from Table 3 (prospective data)?
7
A high degree of tuberculin sensitivity is also considered to be a risk factor for tuberculosis
in many populations. Data on this characteristic from Muscogee County, Georgia can also be
used to compare results of a cross-sectional and a prospective study. As mentioned earlier, a
community-wide screening project was carried out in 1950. In this project, participants were
given a standard tuberculin skin test and a chest photofluorograph. Persons with abnormal
photofluorographs were recalled for a 14x17 radiograph and a clinical examination. Table 4
shows the frequency of pulmonary tuberculosis among the screened population by size of
tuberculin reaction (diameter of induration in mm). Table 5 shows the rate of new tuberculosis
developing during the next 14 years among the population with normal chest radiographs in
1950, according to the size of their tuberculin reaction in 1950.
Table 4
Cases of pulmonary tuberculosis detected in surveyed population by size of tuberculin reaction,
Muscogee County, Georgia, 1950.
Induration
(mm)
No. of
Cases
Cases
per 1000
496
11.8
0- 2
34
4.1
3- 4
38
4.5
1.1
5- 7
71
8.5
2.1
8-12
142
16.9
13+
211
25.2
Total
8
Relative
Risk
1.0
6.1
Table 5
Cases of pulmonary tuberculosis among surveyed population developed during a 14-year period
(1950-1964), by size of tuberculin reaction in 1950, Muscogee County, Georgia.
Induration
(mm)
No. of
Cases
Cases
per 1000
Relative
Risk
Total
239
26.6
0
20
8.7
1.0
1- 4
43
16.8
1.9
5- 9
66
26.9
3.1
10 - 14
72
61.9
15+
38
77.0
8.9
( 3) Complete tables 4 and 5, i.e., calculate the missing relative risks,
using persons with the smallest tuberculin reactions as the reference
group. Note that relative risk is calculated as rate of exposed
divided by rate of unexposed (e.g., 0 mm or 0-2 mm induration). Is
the association of tuberculosis with size of reaction similar in the
cross-sectional and prospective studies?
9
[To assist you with answering the above question, it would be wise to plot the relative risk
values from the two tables onto a single piece of semi-log graph paper. (x-axis = mm induration,
and y-axis = relative risk). Label the graph succinctly but completely.]
( 4) When are associations observed from cross-sectional (prevalence)
studies similar to those observed from prospective studies?
10
PART II: RELATIVE RISK AND RELATIVE ODDS
For this part of the exercise, a defined population has been selected among persons
identified in a private census of Washington County, Maryland, as of 15 July 1963, namely white
males and females who were aged 45 through 64 inclusive on the census date. This subset
includes over 95% of the white population in the county in that age range. Ethnic groups other
than whites have been excluded because there is some evidence that their experience with the
illness of interest (cancer of the colon) is different from that of whites, and there are too few nonwhites in Washington County to allow reliable estimates of their experience. A further simplifying
assumption is that there were no losses (e.g., migration) from the population.
White males and females aged 45 through 64 on 15 July 1963 who could be identified in
the 1963 census lists were the source population for this study. Cases were in the county
cancer register as having had a diagnosis of cancer of the colon first made in the 12-year period
15 July 1963 through 14 July 1975. It is believed that identification of cancer cases is very
nearly complete, but this cannot be known for certain. For the purposes of this exercise,
assume that ascertainment was complete.
( 5) Under what circumstances would incomplete ascertainment affect estimates
of relative risks or relative odds?
Cancer of the colon has been reported to be more common in urban populations than in
rural populations, and more common in high socio-economic groups (as measured by average
education levels in the area in which they reside) than in lower socio-economic groups.
Table 6 shows the information needed to estimate the relative risks of developing colon
cancer by urban residence and high socio-economic status (defined as having completed 13 or
more grades of school). Urban includes Hagerstown suburbs.
11
Table 6
Number and rate per 1000 white residents of Washington County, MD aged 45 through 64 years on 15 July
1963, of cases of cancer of the colon diagnosed 15 July 1963 through 14 July 1975, by residence and grades
of school completed.
Cases
Initial characteristic
Population
Total
18,125
116
Urban
9,351
50
Rural
8,774
66
2,418
23
15,707
93
N
Rate
(per1000)
Relative
risk
95%
Confidence
limits**
Relative
odds
0.71
0.49 - 1.03
0.71
6.4
Residence
5.3
7.5
1.00
1.00
5.9
1.00
1.00
Grades completed
13+
<13, NS*
* NS: not stated.
** 95% confidence limits for the Relative Risk, see formula below.
See lecture handout entitled "Measures of Association" for method of calculating the
confidence limits for the relative risk (Katz et al, Biometrics 34: 469-74, 1978). Briefly, the
variance of the natural log of the relative risk can be approximated as follows (Katz et al. 1978,
Kahn & Sempos, pp. 62-63):2
b
d
VAR (ln(RR)) = a + c
a+b c +d
2 When giving formulas for variance estimates of measures of association for unmatched data in a 2x2
table, the following notation is used in this exercise:
Cases
Controls
a
b
c
d
12
( 6) Complete the blanks in table 6. Do the relative risks and odds ratios differ
from each other to any meaningful degree? Be sure you know under what
conditions they would be expected to be similar and markedly dissimilar.
A sample of controls was selected for the source population who were never identified as
cases. Pertinent information was abstracted from the listing and entered on the computer file in
the same way that had already been done for the 116 cases. Table 7 shows the results.
Table 7
Residence and grades of school completed for cases of cancer of the colon and a sample of
controls, white males and females aged 45 through 64, identified in the 1963 census of
Washington County, MD
Initial characteristic
Cases
Controls
116
116
Urban
50
66
Rural
66
50
23
12
Total
Relative
odds
95%
Confidence
limits
Residence
1.00
Grades completed
13+
<13, NS*
93
* NS: Not stated
( 7) What type of epidemiologic study is this?
13
104
2.14
1.00
1.01 - 4.55
See lecture handout entitled "Measures of Association" for method of calculating these
confidence limits (Woolf B: Ann Hum Gen 10:251-253, 1955). Briefly, the variance of the natural
log of the odds ratio can be approximated as follows (Woolf, 1955, Kahn & Sempos, pp. 56-58):
VAR (ln(RO)) =
1 1 1 1
+ + +
a b c d
where a,b,c, and d are the entries in the 2x2 table (see footnote on page 12).
( 8) What is the total reference population from which the control group was
selected?
( 9) Calculate the relative odds and its 95% confidence limits for residence in
table 7.
(10) Do the results in this table differ from those in Table 6? Why?
In this exercise, you have the unique opportunity of seeing how closely the control odds
(urban/rural; 13+/0-12, NS) reflect the true odds which can be obtained from Table 6. Only
rarely in the real world will you have such an opportunity. In the future, always keep in mind that
your findings may result as much or even more from sampling variation than from any true
association. Even at very low p-values, chance may have produced your findings. Rare events
are happening all the time!
(11) From Table 6, what are the expected numbers of urban and rural controls in
table 7? How might you explain the discrepancy with the number of controls
actually selected (table 7)?
14
As it happened, in the initial selection of controls, one case was included in the 116
persons. When this was noted, searching of the lists was resumed at the point where the case
was located, and the next person who met the criteria for controls and who was not a case was
substituted for them. The control group, therefore, is a sample of non-cases in the population.
If the cases had been retained in the sample, however, the groups would not have been a
sample of non-cases but rather a sample of the total study population. Table 8 shows a sample
of the study population, which in this particular instance happened to include 1 case.
Table 8
Residence and grades of school completed for cases of cancer of the colon and a sample of
white males and females aged 45 through 64 years who were identified in the 1963 census of
Washington County, MD
Initial characteristic
Cases
Sample
116
116
Urban
50
58
Rural
66
58
23
9
93
107
Total
Residence
1.00
Grades completed
13+
<13, NS*
* NS: Not stated
(12) When the cross-products approach used in Table 7 is applied to Table 8,
what measure of association is produced?
15
1.00
When cases are compared in this way with a sample of the population, there are no simple
ways of obtaining confidence limits. If, however, the disease is rare, the same formula used for
confidence limits of relative odds will give limits that are suitable for most practical purposes.
A second set of controls matched to the cases was drawn. For each case, a non-case of
the same race, sex and year of age was selected. The results are shown in Table 9 as counts
of individual controls, not as matched pairs.
Table 9
Residence and grades of school completed for cases of cancer of the colon and randomly
selected controls matched to cases by race, sex and year of age, white males and females aged
45 through 64 identified in the 1963 census of Washington County, MD
Initial characteristic
Total
Cases
Matched
Controls
116
116
Relative odds
Residence
Urban
50
55
Rural
66
61
23
16
93
100
1.00
Grades completed
13+
<13, NS*
1.00
* NS: Not stated
(13) Why do the results in Table 9 differ from those in Tables 6 and 7? Which of
the two control sets is likely to be most appropriate for the assessment of
risk associated with education level attained? Why?
16
While the relative odds calculated in this way (e.g. by pooling matched pairs) yields an
estimate that is likely to be closer to the truth than if matching were not done, it is not a socially
accepted method. Some epidemiologists and biostatisticians even become rather violent in their
objections. With matched controls, a different method of calculating the relative odds should be
employed. Not only is the arithmetic simpler, but the resulting odds ratio is unbiased. The table
is set up as illustrated in Table 10 for residence. Use the data in Table 11 to obtain the numbers
of years of school completed.
Table 10
Numbers of pairs of cases of cancer of the colon and matched controls by residence and
schooling classification of case and control in each pair
Matched
controls
Cases
Relative
odds
95%
Confidence
limits**
0.84
0.50 - 1.40
RESIDENCE
Urban
Rural
Urban
23 (a)
32 (b)
Rural
27 (c)
34 (d)
GRADES COMPLETED
13+
<13, NS*
13+
<13, NS*
* NS: Not stated
** See formula below.
17
Cells "a" and "d" in Table 10, where the residence and schooling histories of cases and
controls in each pair agree, are not considered to add any useful information, and are
consequently disregarded. The relative odds are calculated by dividing the number of pairs in
cell "c" by the number of pairs in cell "b". Note that the total number in all four cells or the two
parts of Table 10 is not the number of subjects but the number of matched pairs. Note also that
when the table is arranged with case characteristics at the top, the matched pairs odds ratio is
c/b, not b/c as in many textbook illustrations where control characteristics are placed at the top.
Moral: Don't assume that everyone labels their tables in the same way. For the confidence
limits, calculate the variance of the log RO as detailed in your lecture handouts:
VAR (ln( ROpaired )) =
1 1
+
b c
(14) What are the advantages and disadvantages of adjustment by using
matched controls? If the value of relative odds calculated by the matched
pair method is identical to the value obtained by the pooled method, what
does this tell you about the usefulness of matching?
The use of matched pairs in this instance would have made it possible to use the method
of sequential analysis to stop the study as soon as a relative risk had been found that was
significantly different from 1.00 (or any other predetermined value) at any predetermined level of
significance. Although sequential analysis is primarily designed for use in therapeutic trials in
which the endpoint for each subject is rather quickly determined, there is no reason why the
procedure might not be applied to matched case-control studies in order to hold the time and
expense of the study to the minimum required for a definitive answer.
In Washington County, sequential analysis has been particularly useful when trying out a
new interview procedure or questionnaire. In such situations, one wishes to know whether the
new represents a significant improvement over the old as quickly as possible. By allocating one
of each matched pair of subjects to one procedure and the other subject to the other procedure,
one can stop the experiment as soon as a significant answer is obtained -- or when enough
pairs have been exposed to the two methods that it is clear that neither has an important
advantage. The method is explained in Armitage P: Sequential Medical Trials, Blackwell, 1960,
and illustrated in an article by Snell and Armitage in Lancet 1:860, 1957. Another useful article
is Bross, I: Sequential medical plans, Biometrics, pp 188-205, Sept. 1952. A few pages from
this article are appended (Appendix 1).
18
Table 11
Years of school completed for cases and their matched controls.
Pair
Case
Cont
Pair
Case
Cont
Pair
Case
Cont
Pair
Case
Cont
1
2
3
4
5
9
10
8
7
14
12
8
99
4
14
30
31
32
33
34
12
8
6
10
14
7
11
16
8
8
59
60
61
62
63
3
8
12
6
14
9
8
8
7
18
88
89
90
91
92
6
8
14
12
8
8
8
6
13
4
6
7
8
9
10
8
15
13
8
11
4
9
99
17
15
35
36
37
38
39
9
8
2
11
7
8
8
8
11
12
64
65
66
67
68
8
7
8
12
8
9
6
7
9
0
93
94
95
96
97
4
12
7
8
8
99
5
12
8
8
11
12
13
14
15
8
11
16
20
12
8
12
10
8
5
40
41
42
43
44
10
8
9
12
6
8
12
16
7
12
69
70
71
72
73
8
9
18
8
16
7
8
14
8
6
98
99
100
101
102
12
13
14
9
12
8
11
8
14
6
16
17
18
19
20
12
11
7
10
12
6
12
2
8
12
45
46
47
48
49
10
8
8
12
12
6
16
10
13
7
74
75
76
77
78
8
14
8
8
12
12
8
3
6
9
103
104
105
106
107
10
6
8
8
12
13
11
8
12
10
21
22
23
24
25
10
5
8
16
17
14
6
4
5
6
50
51
52
53
54
12
8
8
12
8
12
8
3
7
8
79
80
81
82
83
5
15
9
16
8
6
8
12
14
6
108
109
110
111
112
10
9
8
7
8
99
9
15
14
12
26
27
28
29
9
12
12
16
7
12
6
11
55
56
57
58
7
8
14
12
6
5
4
12
84
85
86
87
8
13
12
13
12
11
4
7
113
114
115
116
17
11
16
8
8
5
9
8
19
PART III: A SIMPLE TEST FOR THE SIGNIFICANCE
OF A LINEAR TREND, TREND CHI-SQUARE
Referring again to Part I of this exercise, in 1961, subcutaneous fat thickness was
measured on the photofluorographs of 24,390 persons who participated in the 1946 survey in
Muscogee County, GA (Comstock, Kendrick, Livesay: Subcutaneous fatness and mortality. Am
J Epidemiology 83:548, 1966). Table 12 shows data from this study.
Table 12
Tuberculosis mortality during the period 1946-1961 by thickness of
fat layer over the trapezius ridge in 1946 among black persons
aged 15-34 years in 1946 whose chest photofluorographs showed
no evidence of tuberculosis in 1946.
Fat
Thickness
(mm)
Midpoint
Population
Tuberculosis Deaths
Number
Rate
0-4
1690
14
.00828
5-9
1845
6
.00325
10+
766
2
.00261
Total
4301
22
.00512
If one applies the usual chi-square test with two degrees of freedom to the above table,
one obtains a value of 5.54, equivalent to a p-value of 0.063. Because this p-value falls above
the widely esteemed threshold of 0.05, many persons would shrug off such a finding as
non-significant.
The original hypothesis of the study, however, was that thin people would have higher
tuberculosis death rates than fat people. This hypothesis calls for rates to fall in order, with the
highest rate for the 0-4 mm fat layer group and the lowest rate for the group with fat layers of 10
mm or more. The ordinary Chi-square test gives the likelihood that chance alone could produce
the observed values in any of six different orders. Clearly, the likelihood of getting these rates in
the predicted order purely by chance must be appreciably less than getting them in any of six
orders.
A convenient and widely applicable test for the probability that chance alone could account
for obtaining values in a predicted order is given in Statistical Methods by Snedecor and
Cochran, 7th Edition, 1990, pp. 204-208 (also Cochran WG: Some methods for strengthening
the common χ2 tests. Biometrics 10:434-435, 1954.) The hypothesis to be tested in this case is
that the proportion of tuberculosis deaths decreases with increasing fat thickness. Graphically,
this would correspond to fitting a straight line through the three points. The hypothesis can be
20
tested by calculating the slope3 of the line and testing whether it is significantly different from 0.
The figure below shows the graphical result, and table 13 shows the data presented in
table 12, plus additional columns that have been added to facilitate the calculations needed to
estimate the slope ß. Note that the three fat thickness categories have been treated as ordinal
values and given arbitrary scores -1, 0, and 1 (see below).
Table 13
Work table for the calculation of slope ß (data from table 12).
Fat
Thickness
Midpoint
Population
Tuberculosis Deaths
mm.
(ni)
Number
(ai)
0-4
1690
14
5-9
1845
10+
Total
Rate
(pi)
Score
(Xi)
aiXi
niXi
niXi2
.00828
-1
-14
-1690
1690
6
.00325
0
0
0
0
766
2
.00261
+1
N = 4301
22
p = .00512
3 Slope (usually denoted as ß) is the average change in the ordinate (rate) of the regression function
associated with a unit change in the abscissa (fat thickness).
21
By filling in the work table and using the equations below you can calculate the slope ß,
and its standard deviation, Sß. The hypothesis that ß=0 (flat line) can be tested by calculating z,
the number of standard deviations that the estimated ß is away from the null hypothesis (ß=0).
Z should be normally distributed and its square (z2) should have a chi-square distribution.
(15) Fill in the blanks in Table 13. To calculate ß, it is simplest to consider its
numerator and denominator separately:
NUM = (ai Xi) -
(ai) (ni Xi)
(22) (-924)
= (-12) = - 7.274
N
4301
DEN = (ni Xi2) -
 =
s =
z =
(ni Xi)2
N
= (2456) -
(-924 )2
= 2257.49
4301
NUM
= - 0.00322
DEN
pq
= 0.001502
DEN
 -0
s
= - 2.145 ,
p = 0.032
[Note: ß-0 (beta minus zero) is analogous to O-E (observed minus expected). Beta is the
observed slope and 0 is the expected under the null hypothesis being tested.]
"z" can be converted into chi-square with 1 degree of freedom:
2
2
1 = z2 = (-2.145 ) = 4.601
(16) What is the meaning of the ß value you calculate? For a trend test, what
does a negative value of z signify?
(17) Is a statistically significant trend test synonymous with a significant doseresponse trend?
22
Note that the calculation is greatly simplified by selecting the values of X as -1, 0 and +1. Other
numbers could be used, such as 1, 2, and 3. Or if one had evidence that group 10+ was much
fatter than that order implied, one could choose 1, 2, and 4, if this seemed to be the appropriate
progression of fatness. With continuous variables and modern calculators, it is best to use the
mean value of each category (category of fat thicknesses in this example). These mean values
are 3.1, 7.0 and 14.1 mm.
If an order can be predicted ahead of time, and if reasonable numerical values can be
attached to the ranking orders, this is a more appropriate test than one that does not take order
into account. Snedecor and Cochran state that "moderate differences between two scoring
systems seldom produce marked differences in the conclusions drawn from the analysis". Try
this out for yourself by letting Xi equal 3.1 for the 0-4 mm group, 7.0 for the 5-9 mm group, and
14.1 for the 10+ mm group.
23
Appendix 1
24
Appendix 1
25