Interpreting regression for non-statisticians

Colin Fischbacher

What this presentation will cover

• overview of regression methods• what are they and why use them?• what do results from regression look like?• how do you interpret those results?• what pitfalls should I look out for?

What is regression?

• regression relates two kinds of variables:• outcome variables: for example

– 30 day mortality– Blood pressure– CHD admission rate

• explanatory variables: for example– age– sex– treatment type

What is regression? (2)

are these variables related?

if so, in what way?

What is regression? (3)the red line is an estimate of the relationship that best fits the data we have

other estimates are possible

What is regression? (3)the red line is an estimate of the relationship that best fits the data we have

other estimates are possible

What is regression? (4)

regression can examine more than one explanatory variable at a time

males in red,females in black. . . females have higher blood pressure overall

What is regression? (5)

males in red,

females in black

. . at each age male blood pressure is higher

What is regression? (6)

• here regression is used to estimate how much blood pressure rises with age (so many mm/yr)

• taking this effect of age into account, regression is used to estimate how much higher male blood pressure is than female blood pressure (so many mm higher, taking into account age)

males in red,females in black

Why use regression methods?

• There are other methods to adjust for one or two variables– standardisation– stratification

• These methods deal well with one or two explanatory variables (usually age or sex)

• Regression allows you to take into account the effect of many variables at the same time

• Answers the question “What’s the effect of this variable allowing for all the other ones in the model?”

What methods are available?

• Depends on outcome variable . . • Continuous variable (eg blood pressure)

– linear regression

• Yes/no/binary outcome (eg dead/alive)– logistic regression

• Rate variable (eg admissions per year)– Poisson regression

• Time to event (eg death from cancer)– Cox regression/ survival analysis

• (Many other types also available)

Linear regression Continuous variable (eg blood pressure)

Linear regressionContinuous outcome data (eg blood pressure)

Blood pressure

mmHg

Age (per year) 0.5 (0.3, 0.7)

Sex (male) 4.0 (3.5, 4.5)

Ethnic group

White 0 (ref)

South Asian 3.5 (3.0, 4.0)

Afro-Carribean 4.1 (3.6, 4.6)

Logistic regressionYes/no/binary outcome (eg dead/alive)

Death within 30 days of heart attack

Age Odds ratio (95% CI)

30-50 years 1.0

51-60 years 1.5 (1.1, 1.9)

61-80 years 2.5 (1.5, 3.0)

Sex

Male 1.0

Female 1.2 (1.1, 1.3)

Blood pressure (per 10mmHg) 1.5 (1.4, 1.6)

Poisson regressionRate variable (eg admissions per year)

Emergency admission for COPD

Sex Rate ratio (95% CI)

Females 1.0

Males 1.2 (0.5, 1.9)

Additional co-morbidities

None 1.0

Present 2.5 (2.2, 2.8)

Age (per 10 year increase) 1.5 (1.3, 1.7)

Cox regressionTime to event (eg recurrence of cancer)

Time to recurrence of cancer

Treatment Hazard ratio (95% CI)

Previous treatment 1.0

New drug X 0.5 (0.2, 0.8)

Stage of disease

Grade 1 1.0

Grade 2 0.9 (0.5, 1.3)

Grade 3 1.5 (1.2, 1.8)

Age 1.01 (1.005, 1.015)

Some notes of caution

• Regression is technically easy with most stats packages (point and click)

• However skill is needed:– to choose the right method and the best model– to select how many and which variables to include– to check that the final model fits well– to interpret the final results

• There are always important assumptions• Modelling requires experience and judgement

and includes a degree of subjectivity

What should I look for?

• The kind of model used (logistic, Poisson etc)• The variables included in the model• The effect estimates for each variable (or

“parameter”)• For each categorical variable an indication of

which category is the reference category (usually given a null effect size)

• An assessment of the goodness of model fit

What do the results mean?

Effect estimates (may be called coefficients) may be:• Single figures• Odds ratios• Rate ratios• Hazard ratios

Linear regressionContinuous outcome data (eg blood pressure)

Blood pressure

mmHg (95% CI)

Age (per year) 0.5 (0.3, 0.7)

Sex (male) 4.0 (3.5, 4.5)

Ethnic group

White 0 (ref)

South Asian 3.5 (3.0, 4.0)

Afro-Carribean 4.1 (3.6, 4.6)

Logistic regressionYes/no/binary outcome (eg dead/alive)

Death within 30 days of heart attack

Age Odds ratio (95% CI)

30-50 years 1.0

51-60 years 1.5 (1.1, 1.9)

61-80 years 2.5 (1.5, 3.0)

Sex

Male 1.0

Female 1.2 (1.1, 1.3)

Blood pressure (per 10mmHg) 1.5 (1.4, 1.6)

Poisson regressionRate variable (eg admissions per year)

Emergency admission for COPD

Sex Rate ratio (95% CI)

Females 1.0

Males 1.2 (0.5, 1.9)

Additional co-morbidities

None 1.0

Present 2.5 (2.2, 2.8)

Age 1.01 (1.005, 1.015)

Cox regressionTime to event (eg recurrence of cancer)

Time to recurrence of cancer

Treatment Hazard ratio (95% CI)

Previous treatment 1.0

New drug X 0.5 (0.2, 0.8)

Stage of disease

Grade 1 1.0

Grade 2 0.9 (0.5, 1.3)

Grade 3 1.5 (1.2, 1.8)

Age (per 10 years) 1.5 (1.4, 1.6)

What else should I look for?

• Is the basic question clear?– why was a regression method chosen?

• Was the correct model used?– logistic if yes/no outcomes, Poisson if rates etc

• Which variables were included?– Were any ones you think are important left out?

• How were the variables chosen?– modelling strategies and results of exploration?

• How many variables were included?– 10 -20 cases per variable approximate rule of thumb

• Effect sizes (or “coefficients”) and confidence intervals• Were measures of model fit reported?

REAL LIFE EXAMPLESregression methods

Cox regressionMcBride and colleagues (BMJ Dec 4, 2010) conducted a study of patients in 324 UK general practices and examined the time they waited between consulting their GP with hip pain and being referred to secondary care.

The figures show hazard ratios for referral from a Cox regression model that included age group, sex and deprivation quintile

Poisson regressionSim and colleagues (BMJ Dec 4, 2010) conducted a study to examine changes in the rate of emergency admission for acute myocardial infarction before and after the introduction of smoke free legislation in England. After adjusting for year of admission, temperature, Christmas holidays and week of admission in a Poisson regression model, they obtained the results shown in the table.

BMJ 340: doi:10.1136/bmj.c2161

Logistic regressionAlm and colleagues interviewed parents of 294 cases of Sudden Infant Death Syndrome (SIDS) in three Scandinavian countries, asking about coffee and alcohol consumption by the mother.

* adjusted for maternal smoking in 1st trimester, maternal age, education and parity

Arch Dis Child 1999;81:107-111 doi:10.1136/adc.81.2.107

Conclusions

Regression methods allow you to examine the effects of many variables simultaneously

However they do not give “automatic” answers

Care is needed in choice of method, selection of variables, testing the final model and interpreting the results

Model building always involves some degree of judgement and personal choice