Goals
Understand confidence intervals and p-values
Learn to use basic statistical tests including chi square and ANOVA
Types of Variables Types of variables indicate which estimates you
can calculate and which statistical tests you should use
Continuous variables: Always numeric Generally calculate measures such as the mean,
median and standard deviation Categorical variables:
Information that can be sorted into categories Field investigation – often interested in dichotomous or
binary (2-level) categorical variables Cannot calculate mean or median but can calculate
risk
Measures of Association Strength of the association between two
variables, such as an exposure and a disease Two measure of association used most often
are the relative risk, or risk ratio (RR), and the odds ratio (OR)
The decision to calculate an RR or an OR depends on the study design
Interpretation of RR and OR: RR or OR = 1: exposure has no association with
disease RR or OR > 1: exposure may be positively associated
with disease RR or OR < 1: exposure may be negatively associated
with disease
Risk Ratio or Odds Ratio? Risk ratio
Used when comparing outcomes of those who were exposed to something to those who were not exposed
Calculated in cohort studies Cannot be calculated in case-control studies because
the entire population at risk is not included in the study
Odds ratio Used in case-control studies Odds of exposure among cases divided by odds of
exposure among controls Provides a rough estimate of the risk ratio
Analysis Tool: 2x2 Table
Commonly used with dichotomous variables to compare groups of people
Table puts one dichotomous variable across the rows and another dichotomous variable along the columns
Useful in determining the association between a dichotomous exposure and a dichotomous outcome
Calculating an Odds Ratio
Table displays data from a case control study conducted in Pennsylvania in 2003 (2)
Can calculate the odds ratio: *OR = ad = (218)(85) = 19.6
bc (45)(21)
Outcome
Exposure
Hepatitis ANo
Hepatitis ATotal
Ate salsa 218 45 263
Did not eat salsa
21 85 106
Total 239 130 369
Table 1. Sample 2x2 table for Hepatitis A at Restaurant A
Confidence Intervals
Point estimate – a calculated estimate (like risk or odds) or measure of association (risk ratio or odds ratio)
The confidence interval (CI) of a point estimate describes the precision of the estimate The CI represents a range of values on
either side of the estimate The narrower the CI, the more precise the
point estimate (3)
Confidence Intervals - Example
Example—large bag of 500 red, green and blue marbles: You want to know the percentage of green
marbles but don’t want to count every marble Shake up the bag and select 50 marbles to
give an estimate of the percentage of green marbles
Sample of 50 marbles: 15 green marbles, 10 red marbles, 25 blue marbles
Confidence Intervals - Example
Marble example continued: Based on sample we conclude 30% (15 out
of 50) marbles are green 30% = point estimate
How confident are we in this estimate? Actual percentage of green marbles could
be higher or lower, ie. sample of 50 may not reflect distribution in entire bag of marbles
Can calculate a confidence interval to determine the degree of uncertainty
Calculating Confidence Intervals
How do you calculate a confidence interval?
Can do so by hand or use a statistical program Epi Info, SAS, STATA, SPSS and Episheet are
common statistical programs Default is usually 95% confidence
interval but this can be adjusted to 90%, 99% or any other level
Confidence Intervals Most commonly used confidence interval is the
95% interval 95% CI indicates that our estimated range has a 95%
chance of containing the true population value Assume that the 95% CI for our bag of marbles
example is 17-43% We estimated that 30% of the marbles are
green: CI tells us that the true percentage of green marbles
is most likely between 17 and 43% There is a 5% chance that this range (17-43%) does
not contain the true percentage of green marbles
Confidence Intervals
If we want less chance of error we could calculate a 99% confidence interval A 99% CI will have only a 1% chance of
error but will have a wider range 99% CI for green marbles is 13-47%
If a higher chance of error is acceptable we could calculate a 90% confidence interval 90% CI for green marbles is 19-41%
Confidence Intervals Very narrow confidence intervals indicate a very
precise estimate Can get a more precise estimate by taking a
larger sample 100 marble sample with 30 green marbles
Point estimate stays the same (30%) 95% confidence interval is 21-39% (rather than 17-43%
for original sample) 200 marble sample with 60 green marbles
Point estimate is 30% 95% confidence interval is 24-36%
CI becomes narrower as the sample size increases
Confidence Intervals Returning to example of Hepatitis A in a
Pennsylvania restaurant: Odds ratio = 19.6 95% confidence interval of 11.0-34.9 (95% chance
that the range 11.0-34.9 contained the true OR) Lower bound of CI in this example is 11.0 (e.g., >1)
Odds ratio of 1 means there is no difference between the two groups, OR > 1 indicates a greater risk among the exposed
Conclusion: people who ate salsa were truly more likely to become ill than those who did not eat salsa
Confidence Intervals Must include CIs with your point estimates to
give a sense of the precision of your estimates Examples:
Outbreak of gastrointestinal illness at 2 primary schools in Italy (4)
Children who ate corn/tuna salad had 6.19 times the risk of becoming ill as children who did not eat salad
95% confidence interval: 4.81 – 7.98 Pertussis outbreak in Oregon (5)
Case-patients had 6.4 times the odds of living with a 6-10 year-old child than controls
95% confidence interval: 1.8 – 23.4 Conclusion: true association between exposure and
disease in both examples
Analysis of Categorical Data
Measure of association (risk ratio or odds ratio)
Confidence interval Chi-square test
A formal statistical test to determine whether results are statistically significant
Chi-Square Statistics
A common analysis is whether Disease X occurs as much among people in Group A as it does among people in Group B People are often sorted into groups based
on their exposure to some disease risk factor
We then perform a test of the association between exposure and disease in the two groups
Chi-Square Test: Example
Hypothetical outbreak of Salmonella on a cruise ship Retrospective cohort study conducted All 300 people on cruise ship
interviewed, 60 had symptoms consistent with Salmonella
Questionnaires indicate many of the case-patients ate tomatoes from the salad bar
Chi-Square Test: Example (cont.)
To see if there is a statistical difference in the amount of illness between those who ate tomatoes (41/130) and those who did not (19/170) we could conduct a chi-square test
Salmonella?
Yes No Total
Tomatoes 41 89 130
No Tomatoes 19 151 170
Total 60 240 300
Table 2a. Cohort study: Exposure to tomatoes and Salmonella infection
Chi-Square Test: Example (cont.)
To conduct a chi-square the following conditions must be met: There must be at least a total of 30
observations (people) in the table Each cell must contain a count of 5 or more
To conduct a chi-square test we compare the observed data (from study results) with the data we would expect to see
Chi-Square Test: Example (cont.)
Salmonella?
Yes No Total
Tomatoes 130
No Tomatoes 170
Total 60 240 300
Gives an overall distribution of people who ate tomatoes and became sick
Based on these distributions we can fill in the empty cells with the expected values
Table 2b. Row and column totals for tomatoes and Salmonella infection
Chi-Square Test: Example (cont.)
Expected Value = Row Total x Column Total
Grand Total
For the first cell, people who ate tomatoes and became ill:
Expected value = 130 x 60 = 26 300 Same formula can be used to calculate the
expected values for each of the cells
Chi-Square Test: Example (cont.)
Salmonella?
Yes No Total
Tomatoes130 x 60 = 26
300 130 x 240 = 104 300
130
No Tomatoes170 x 60 = 34
300 170 x 240 = 136 300
170
Total 60 240 300
To calculate the chi-square statistic you use the observed values from Table 2a and the expected values from Table 2c
Formula is [(Observed – Expected)2/Expected] for each cell of the table
Table 2c. Expected values for exposure to tomatoes
Chi-Square Test: Example (cont.)
Salmonella?
Yes No Total
Tomatoes(41-26)2 = 8.7
26
(89-104)2 = 2.2 104
130
No Tomatoes(19-34)2 = 6.6
34
(151-136)2 = 1.7 136
170
Total 60 240 300
The chi-square (χ2) for this example is 19.2 8.7 + 2.2 + 6.6 + 1.7 = 19.2
Table 2d. Expected values for exposure to tomatoes
Chi-Square Test
What does the chi-square tell you? In general, the higher the chi-square
value, the greater the likelihood there is a statistically significant difference between the two groups you are comparing
To know for sure, you need to look up the p-value in a chi-square table
We will discuss p-values after a discussion of different types of chi-square tests
Types of Chi-Square Tests
Many computer programs give different types of chi-square tests
Each test is best suited to certain situations
Most commonly calculated chi-square test is Pearson’s chi-square Use Pearson’s chi-square for a fairly
large sample (>100)
Types of Statistical Tests
Parade ofStatistics Guys
The right test...
To use when….
Pearson chi-square (uncorrected) Sample size >100Expected cell counts > 10
Yates chi-square (corrected) Sample size >30Expected cell counts ≥ 5
Mantel-Haenszel chi-square Sample size > 30Variables are ordinal
Fisher’s exact test Sample size < 30 and/orExpected cell counts < 5
Using Statistical Tests:Examples from Actual Studies
In each study, investigators chose the type of test that best applied to the situation (Note: while the chi-square value is used to determine the corresponding p-value, often only the p-value is reported.)
Pearson (Uncorrected) Chi-Square : A North Carolina study investigated 955 individuals because they were identified as partners of someone who tested positive for HIV. The study found that the proportion of partners who got tested for HIV differed significantly by race/ethnicity (p-value <0.001). The study also found that HIV-positive rates did not differ by race/ethnicity among the 610 who were tested (p = 0.4). (6)
Using Statistical Tests:Examples from Actual Studies
Additional examples: Yates (Corrected) Chi-Square: In an outbreak of
Salmonella gastroenteritis associated with eating at a restaurant, 14 of 15 ill patrons studied had eaten the Caesar salad, while 0 of 11 well patrons had eaten the salad (p-value <0.01). The dressing on the salad was made from raw eggs that were probably contaminated with Salmonella. (7)
Fisher’s Exact Test: A study of Group A Streptococcus (GAS) among children attending daycare found that 7 of 11 children who spent 30 or more hours per week in daycare had laboratory-confirmed GAS, while 0 of 4 children spending less than 30 hours per week in daycare had GAS (p-value <0.01). (8)
P-Values Using our hypothetical cruise ship
Salmonella outbreak: 32% of people who ate tomatoes got
Salmonella as compared with 11% of people who did not eat tomatoes
How do we know whether the difference between 32% and 11% is a “real” difference? In other words, how do we know that our chi-
square value (calculated as 19.2) indicates a statistically significant difference?
The p-value is our indicator
P-Values
Many statistical tests give both a numeric result (e.g. a chi-square value) and a p-value
The p-value ranges between 0 and 1 What does the p-value tell you?
The p-value is the probability of getting the result you got, assuming that the two groups you are comparing are actually the same
P-Values Start by assuming there is no difference in
outcomes between the groups Look at the test statistic and p-value to see if
they indicate otherwise A low p-value means that (assuming the groups are
the same) the probability of observing these results by chance is very small
Difference between the two groups is statistically significant
A high p-value means that the two groups were not that different
A p-value of 1 means that there was no difference between the two groups
P-Values
Generally, if the p-value is less than 0.05, the difference observed is considered statistically significant, ie. the difference did not happen by chance
You may use a number of statistical tests to obtain the p-value Test used depends on type of data
you have
Chi-Squares and P-Values If the chi-square statistic is small, the observed
and expected data were not very different and the p-value will be large
If the chi-square statistic is large, this generally means the p-value is small, and the difference could be statistically significant
Example: Outbreak of E. coli O157:H7 associated with swimming in a lake (1)
Case-patients much more likely than controls to have taken lake water in their mouth (p-value =0.002) and swallowed lake water (p-value =0.002)
Because p-values were each less than 0.05, both exposures were considered statistically significant risk factors
Note: Assumptions Statistical tests such as the chi-square assume
that the observations are independent Independence: value of one observation does not
influence value of another If this assumption is not true, you may not use
the chi-square test Do not use chi-square tests with:
Repeat observations of the same group of people (e.g. pre- and post-tests)
Matched pair designs in which cases and controls are matched on variables such as sex and age
Analysis of Continuous Data
Data do not always fit into discrete categories
Continuous numeric data may be of interest in a field investigation such as: Clinical symptoms between groups of patients Average age of patients compared to average
age of non-patients Respiratory rate of those exposed to a
chemical vs. respiratory rate of those who were not exposed
ANOVA
May compare continuous data through the Analysis Of Variance (ANOVA) test
Most statistical software programs will calculate ANOVA Output varies slightly in different programs For example, using Epi Info software:
Generates 3 pieces of information: ANOVA results, Bartlett’s test and Kruskal-Wallis test
ANOVA When comparing continuous variables between
groups of study subjects: Use a t-test for comparing 2 groups Use an f-test for comparing 3 or more groups Both tests result in a p-value
ANOVA uses either the t-test or the f-test Example: testing age differences between 2
groups If groups have similar average ages and a similar
distribution of age values, t-statistic will be small and the p-value will not be significant
If average ages of 2 groups are different, t-statistic will be larger and p-value will be smaller (p-value <0.05 indicates two groups have significantly different ages)
ANOVA and Bartlett’s Test Critical assumption with t-tests and f-tests:
groups have similar variances (e.g., “spread” of age values)
As part of the ANOVA analysis, software conducts a separate test to compare variances: Bartlett’s test for equality of variance
Bartlett’s test: Produces a p-value If Bartlett’s p-value >0.05, (not significant) OK to use
ANOVA results Bartlett’s p-value <0.05, variances in the groups are
NOT the same and you cannot use the ANOVA results
Kruskal-Wallis Test Kruskal-Wallis test: generated by Epi
Info software Used only if Bartlett’s test reveals variances
dissimilar enough so that you can’t use ANOVA
Does not make assumptions about variance, examines the distribution of values within each group
Generates a p-value If p-value >0.05 there is not a significant
difference between groups If p-value < 0.05 there is a significant difference
between groups
Analysis of Continuous DataFigure 1. Decision tree for analysis of continuous data.
Bartlett’s test for equality of variancep-value >0.05?
YES NO
Use ANOVA test
Use Kruskal-Wallis test test
p<0.05 p>0.05 p<0.05 p>0.05
Difference between groups is statistically significant
Difference between groups is statistically significant
Difference between groups is NOT statistically significant
Difference between groups is NOT statistically significant
Conclusion
In field epidemiology a few calculations and tests make up the core of analytic methods
Learning these methods will provide a good set of field epidemiology skills. Confidence intervals, p-values, chi-square
tests, ANOVA and their interpretations Further data analysis may require methods
to control for confounding including matching and logistic regression
References
1. Bruce MG, Curtis MB, Payne MM, et al. Lake-associated outbreak of Escherichia coli O157:H7 in Clark County, Washington, August 1999. Arch Pediatr Adolesc Med. 2003;157:1016-1021.
2. Wheeler C, Vogt TM, Armstrong GL, et al. An outbreak of hepatitis A associated with green onions. N Engl J Med. 2005;353:890-897.
3. Gregg MB. Field Epidemiology. 2nd ed. New York, NY: Oxford University Press; 2002.
4. Aureli P, Fiorucci GC, Caroli D, et al. An outbreak of febrile gastroenteritis associated with corn contaminated by Listeria monocytogenes. N Engl J Med. 2000;342:1236-1241.
References
5. Schafer S, Gillette H, Hedberg K, Cieslak P. A community-wide pertussis outbreak: an argument for universal booster vaccination. Arch Intern Med. 2006;166:1317-1321.
6. Centers for Disease Control and Prevention. Partner counseling and referral services to identify persons with undiagnosed HIV --- North Carolina, 2001. MMWR Morb Mort Wkly Rep.2003;52:1181-1184.
7. Centers for Disease Control and Prevention. Outbreak of Salmonella Enteritidis infection associated with consumption of raw shell eggs, 1991. MMWR Morb Mort Wkly Rep. 1992;41:369-372.
8. Centers for Disease Control and Prevention. Outbreak of invasive group A streptococcus associated with varicella in a childcare center -- Boston, Massachusetts, 1997. MMWR Morb Mort Wkly Rep. 1997;46:944-948.