FrequencyMeasures

Frequency Measures and Other Math!: 

Frequency Measures and Other Math! Neely Kaydos-Daniels, PhD, MSPH

Slide2: 

“One’s knowledge of science begins when he can measure what he is speaking about and express it in numbers” Lord Kelvin (1824-1907)

Why do we need to do this?: 

Why do we need to do this? Measure disease (and health) in different ways Comparisons! Between groups: Nations, states, regions, counties Gender, race Different foods, different experiences

So why can’t we just do this with numbers of cases?: 

So why can’t we just do this with numbers of cases?

Can we compare?: 

Can we compare? 30 cases in Maryland < 34 cases in Washington, DC 165 in Texas ~ 169 in Missouri ~ 182 in Mississippi 46 in Minnesota ~ 45 in Wisconsin 37 in South Dakota ~ 35 in Georgia

Probably not.: 

Probably not. The numbers of cases may be similar, but the number of people in each state is not We need Rates, Ratios, and Proportions!

Objectives: 

Objectives Define and calculate ratios, proportions, and rates, including incidence, prevalence, and attack rates Define risk ratios and odds ratios, and confidence intervals and p-values Know the differences between these measures and the correct way to interpret them

To back up a little…: 

To back up a little… Variable: a way to describe an observation For example: Observation is a person – Me Describe me how? Gender Age Neighborhood Did I eat a certain food at a restaurant?

Line Listing: 

Line Listing

Observations: 

Observations

Variables: 

Variables

Types of Variables: 

Types of Variables Numeric scale = number Age, weight, # of children Ordinal scale = values in graded order Small, medium, large Nominal scale = names of classes Race, make of car, gender, state

Variables: 

Variables

Nominal scale variables: 

Nominal scale variables Lots with only 2 categories (makes our job easier) Male/female Alive/dead Exposed/unexposed Sick/well YES/NO Called dichotomous variables

Fascinating, but why do we care?: 

Fascinating, but why do we care? Frequently use rates, ratios, and proportions to describe dichotomous variables Ratio of men to women Rate of illness in a population Proportion of population that are pet owners

Ratios: 

Ratios Definition: An expression of the relationship between two quantities May be related or totally independent of each other R = (x/y)(10n) R > 0, may or may not have units

Ratio example: 

Ratio example Deer hunters’ take for the year: x Deer with antlers 2 = 0.5 y Deer without antlers 4 OR x Deer with antlers 2 = 0.33 y All deer 6

Proportion: 

Proportion Ratio in which x is included in y From the previous example: Deer with antlers 2 = 0.33 All deer 6

Rate: 

Rate A measure of the frequency with which an event occurs in a defined population over time Properties: Time, place, and population are specified (sometimes implicitly)

Calculating Rate: 

Calculating Rate R = # events in specified period X10n Average population at risk in period

Slide22: 

For example: 350 deer tagged and followed for 2 years. 37 batteries die after 1 year. 73 deer are killed over the 2 years. Numerator = 73 Denominator= (313x2) + (37x1) = 663 Deer death rate is 73/663=0.11 So 11 deer killed per 100 deer per year

More rates: 

More rates Lots of rates have time implied: Mortality rate in WV = # deaths in WV (per year) Average population of WV

Caution!: 

Caution! Some numbers are called rates, but really aren’t Case-Fatality Rate CFR = # persons who die of an illness total # who have the illness

Incidence: 

Incidence Number of new events occurring in a defined population during a specified period of time New events Measures risk

Incidence Rates: 

Incidence Rates Most common way of measuring and comparing the frequency of disease in a population Expresses the risk of illness in a population over time

To calculate incidence rate: 

To calculate incidence rate new cases occurring over a given IR = time period population at risk during the same time period result X 10n

Example: 

Example x = Number of trauma-injury deaths in US in 1998 = 52,506 y = Population at risk in US in 1998 = 270,649,000 n = 5 IR = (x/y) = 0.000194 Multiply by 105 = 19.4 IR = 19.4 per 100,000 population in 1998

Risk Groups: 

Risk Groups Comparing incidence rates in 2 populations – identify high risk groups

Incidence Rates of WNV: 

Incidence Rates of WNV

Attack Rate: 

Attack Rate AR = # of people who became ill x10n # of people at risk People at risk could be those at a party, in a class, on a cruise ship… Usually express AR as a %, so n = 2

Secondary Attack Rate: 

Secondary Attack Rate Measure of frequency of new cases among contacts of known cases # cases among contacts of primary cases SAR = during the period total # of contacts

The calculations: 

The calculations # of children in day care center = 70 # of ill children at day care center = 7 # of contacts (of those 7) at home = 25 # of ill contacts = 5 AR = 7/70 = 0.1 x 100 = 10% AR SAR = 5/25 = 0.2 x100 = 20% SAR

Prevalence: 

Prevalence Number of existing cases (or number with an attribute) during a period of time Old and new cases Period prevalence Point prevalence – snapshot in time

Slide39: 

Incidence Period prevalence Point prevalence

Risk Ratio = Relative Risk: 

Risk Ratio = Relative Risk Compare the risk of a health event between 2 groups RR= risk in population A risk in population B

Example: 

Example Cruise ship A has 2000 passengers, 295 of whom were ill Cruise ship B has 1500 passengers, 108 of whom were ill Risk on ship A = 295/2000 = 0.1475 Risk on ship B = 108/1500 = 0.072 RR = risk on ship A 0.1475 = 2.0 risk on ship B 0.072 The risk of illness among passengers on ship A was 2.0 times the risk of illness among passengers on ship B

Cohort study: 

Cohort study Pick people for participation based on exposure status Followed through time to see if they get ill Can take a long time and be very $$$$ Good for rare exposures, outbreaks possible Retrospective cohort Use Risk Ratios

Case-control study: 

Case-control study Pick people for participation after they have illness Pick ill people as cases, well ones as controls, and compare the groups Faster, good for rare diseases Common for outbreaks Use Odds Ratios (!)

Probability: 

Probability Different ways 1 die can fall Ways to roll the number 2 with 1 die Probability: # ways to roll 2 # of possible ways This is a proportion!

Example: 

Example # of ways a die can fall = 6 # of ways to roll the number 2 = 1 Probability of rolling the number 2 if I roll a die one time: 1 = (a proportion) = Probability = 0.17 6

Odds: 

Odds Odds: Probability that an event occurs divided by the probability that the event doesn’t occur Odds of picking the ace of spades from a deck of cards

Calculation: 

Calculation Probability of picking the ace of spades = 1/52 Probability of not picking the ace of spades = 51/52 Odds = 1/52 = 1 = 0.02 51/52 51

Odds Ratio: 

Odds Ratio A measure to compare 2 groups The odds of a health event happening in population A compared to the odds of the health event happening in population B In case-control studies, odds ratios are used to compare populations

Example: 

Example Outbreak of Norwalk-like virus on a cruise ship Trying to determine if the chicken salad is the source – so we will do a case-control study Pick 2 groups Those who got sick Those who didn’t Look to see if individuals ate chicken salad during the incubation period

Slide51: 

Of 100 people who were ill, 84 ate chicken salad (so 16 did not eat chicken salad) Of 100 well people, 22 ate chicken salad (so 78 did not eat chicken salad) Odds of eating chicken salad among ill people = 84/100 = 84 = 5.25 16/100 16 Odds of eating chicken salad among well people = 22/100 = 22 = 0.28 78/100 78

Slide52: 

OR = odds among ill of eating chicken salad odds among well of eating chicken salad OR = 5.25 = 18.75 0.28 Those who were ill were 18 times more likely to have eaten chicken salad than those who did not get ill

Interpreting Measures of Association: 

Interpreting Measures of Association Odds ratios and risk ratios measure the degree of relatedness of an exposure and a health event (an outcome) Generally, the exposure of interest is in the numerator – but not required The farther away the OR/RR is from 1, the more we would say the exposure and outcome are associated

Back to chicken salad: 

Back to chicken salad OR = odds of chicken salad among ill odds of chicken salad among well If OR(RR) > 1, then chicken salad is a risk factor for being ill If OR < 1, then chicken salad is protective of illness If OR = 1, then there is no association between chicken salad and illness OR can NEVER be negative!

How good are these measures?: 

How good are these measures? It is possible that chicken salad was not the contaminated food item Random chance is always possible Chance can affect population measures We need to determine how likely it is that the reason for our result is chance If a true association, the result generated may be different from the truth How precise is our result?

Precision vs. Accuracy: 

Precision vs. Accuracy Accuracy – is there a true association or not? Precision – how close are we to the true value of the association?

Slide57: 

Poor accuracy, poor precision Good accuracy, poor precision Poor accuracy, good precision Good accuracy and precision

P-value: 

P-value Used in statistical tests Used in LOTS of statistical tests A guide to tell us that a result is “significant” Generally at the 95% level p < 0.05 = Significant at the 95% level = there is a 95% probability that the result is accurate (not by chance)

95% Confidence Intervals: 

95% Confidence Intervals Used with risk ratios and odds ratios Tell us about both precision and accuracy With an OR or RR we have estimated the magnitude of the association – 95% confidence intervals tell us that we can be 95% sure that the true association is somewhere in that interval

Confidence Intervals: 

Confidence Intervals OR Lower 95% CI Upper 95% CI

Interpretation of 95%CI: 

Interpretation of 95%CI Of 20 people who were ill, 8 ate chicken salad (so 12 did not eat chicken salad) Of 20 well people, 3 ate chicken salad (so 17 did not eat chicken salad) OR = 0.67/.176 = 3.8 95%CI = (0.8, 17.2) How do we interpret this result?

Effect of Sample Size: 

Effect of Sample Size Of 200 people who were ill, 80 ate chicken salad (so 120 did not eat chicken salad) Of 200 well people, 30 ate chicken salad (so 170 did not eat chicken salad) OR = 0.67/.176 = 3.8 - The same! 95%CI = (2.3, 6.1) How do we interpret this result?

Conclusion: 

Conclusion When using measures of association – Odds Ratios and Risk Ratios – the 95% confidence intervals are as important as the point estimate! Principles of Epidemiology, Chapters 2 and 3

Objectives: 

Objectives Define and calculate ratios, proportions, and rates, including incidence, prevalence, and attack rates Define risk ratios and odds ratios, and confidence intervals and p-values Know the differences between these measures and the correct way to interpret them