inference - vumc

Inference

What is a population?

What is a population?

A population is the complete set of patients (or subjects or observations) that we hope to learn about.

Examples: Mother-infant pairs, patients with a previous hernia repair, black or white patients on dialysis

Why not collect data from an entire population?

Why not collect data from an entire population?

• $$$$• Impossible

In order to learn about a population, we collect data on a subset of the population. The subset is a sample.

Selecting and collecting data from a subset of the population is sampling.

A few ways one may sample a population+ Random sample+ Systematic sample+ Stratified sample+ Convenience sample

Remember:Before one can sample, one must clearly define the population.

What is the consequence of collecting data from a subset instead of collecting data from the full population?

What is the consequence of collecting data from a subset instead of collecting data from the full population?

Sampling error or estimation error or variation or uncertainty or loss of precision

Embrace uncertainty

Communicate the degree of uncertainty with conclusions drawn from samples.

Examples: confidence intervals, probabilities, interquartile range

How do we learn from a sample?

Population and Sample

A parameter is a numerical measurement that describes a characteristic

of a population

A statistic is a numerical measurement that describes a characteristic of

a sample

In general, we will use a statistic to infer something about a parameter


• Previously in the NHANES study, we computed summary statistics such as sample means and variances or sample proportions to describe our sample.

• Now, not only do we want to describe the sample, we want to learn something about the population.

• Learn about population = inference about population


• What might we learn about a population?

▫ Population mean (μ): Average value assumed by a random variable, also called the expected value.

▫ Population variance (σ2): Variability of the random variable. May also use Population standard deviation (σ).

▫ Other population parameters: median, 25th- or 75th-quantile

Statistical InferenceStatistical inference consists of:

• Estimation: Use sample to estimate population parameter(s) of interest.

▫ Proportion of low birthweight infants.

• Hypothesis testing: Use sample to evaluate population parameter(s).

▫ For now, we will concentrate on estimation.

Statistical Inference

• Estimation consists of point estimates and interval estimates:

▫ Point estimate: A “best guess” of the population parameter based on the sample.

MeanY

Best guess for the mean

Statistical Inference

• Estimation consists of point estimates and interval estimates:

▫ Point estimate: A “best guess” of the population parameter based on the sample.

▫ Interval estimate: A range of “reasonable values”, accounting for sampling variability of the point estimate.

• IMPORTANT: The sample must be representative of the population of interest.

Y

Range of reasonable values for the mean

Example:

Inadvertent enterotomy (IE) occurs during abdominal repair

(like hernia repairs) when an incision is unintentionally made in

the intestine.

Example (cont):

Suppose: In a cohort of 3000 laparoscopic ventral hernia repairs, there were 120 IE. Among 2000 open ventral hernia repairs, there were 40 IE.

What is the estimated proportion (and 95% CI) in each group?

What is the difference in IE proportions (and 95% CI) between laparoscopic and open?

So what?Why does one care about the

difference in proportions and 95% CI?

Confidence Intervals &Inference about the PopulationA confidence interval is a type of inference about the larger population. The interval represents the set of population parameters supported by the data.

Inference by CI - population mean

Example: What is the mean birthweight (and 95% CI) of infants

for mothers that do not smoke, calculated from the Bayside

hospital data?

Inference by CI - population mean

Example (cont):

bwt 115 3054.957 70.1625 2915.965 3193.948 Variable Obs Mean Std. Err. [95% Conf. Interval]

. ci means bwt if smoke == 0

Inference by CI - population proportion

Example: What is the proportion and 95% CI of mothers that

smoke? (Respond on Top Hat.)

Inference by CI - population proportion

Example (cont):

smoke 189 .3915344 .0355036 .324772 .4626181 Variable Obs Proportion Std. Err. [95% Conf. Interval] Wilson

. ci proportions smoke, wilson

Inference by CI – population rate

Example: What is the rate of death and 95% CI in the placebo

population in the Primary Biliary Cirrhosis Trial (liver.dta)?

Inference by CI – population rate

Example (cont):

status 3.06523 19.57439 2.527043 14.93732 25.19612 Variable Exposure Mean Std. Err. [95% Conf. Interval] Poisson Exact

. ci means status if tx == 0, poisson exposure(ot100k)

. gen ot100k = obstime /100000

. use liver.dta

Inference by CI – difference in population means

Example: Difference and CI of mean birthweight between

mothers do and do not smoke (lowbwt.dta).

Inference by CI – difference in population means

Example (cont):

diff 281.7133 103.9741 76.46677 486.9598 combined 189 2944.656 53.02858 729.0224 2840.049 3049.264 smoker 74 2773.243 76.73218 660.0752 2620.316 2926.17nonsmoke 115 3054.957 70.1625 752.409 2915.965 3193.948 Group Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] Two-sample t test with unequal variances

. ttest bwt, by(smoke) unequal

Inference by CI – difference in population proportions

See example of inadvertent enterotomy above.

One can also make inference about the population with hypothesis

testingThere is a connection with confidence intervals.

Generally, studies are designed to identify

Conclusive differences

Conclusive similarities

or

Because of noise or error, a study may also generate

Conclusive differences

Conclusive similarities

or

Inconclusive results

or

To discover conclusive similarities

• A study will a priori establish an equivalence threshold.


• A study will a priori establish an equivalence threshold.

Other related terms:• Null region• Region of practical

equivalence


• A conclusive similarity is identified when the confidence

interval for the difference falls within the equivalence

threshold.

To discover conclusive differences

• A conclusive difference is identified when the confidence

interval for the difference falls outside the equivalence

threshold.

An inconclusive result

• An inconclusive result occurs when the confidence interval

for the difference straddles the equivalence threshold.

Same framework applies to ratios

Conclusive diff

Conclusive similarityInconclusive

Example

Suppose surgeons establish that the rate of surgical site

infections is equivalent for robotic and laparoscopic hernia

repair.

Example

Suppose surgeons establish that the rate of surgical site infections is equivalent for robotic and laparoscopic hernia repair. The surgeons decide on a 2.5 percentage point difference as threshold.

Example

If 89 in a cohort of 1000 laparoscopic repairs and 97 in a cohort

of 1000 robotic repairs experience an infection, then …

Example

If 89 in a cohort of 1000 laparoscopic repairs and 97 in a cohort

of 1000 robotic repairs experience an infection, then …

Inconclusive-1.7 3.3

Example

However, if 892 in a cohort of 10000 laparoscopic repairs and 967 in a cohort of 10000 robotic repairs experience an infection, then …

Example

However, if 892 in a cohort of 10000 laparoscopic repairs and

967 in a cohort of 10000 robotic repairs experience an

infection, then …

Conclusive similarity-0.0 1.6

It is common to have a point null instead of a null region


Conclusive differenceInconclusive


Conclusive differenceInconclusive

Notice that there is no possibility of identifying a conclusive similarity.

A common mistake is to interpret an inconclusive result as evidence for a conclusive similarity.

Inconclusive

You might see in a manuscript: The rates of adverse events were the same in the intervention and placebo groups.


Why can this be misleading?• There usually is no threshold for equivalence• The statement is usually based on a point null• Fails to communicate the degree of differences supported by the data



Why can this be misleading?• There usually is no threshold for equivalence• The statement is usually based on a point null• Fails to communicate the degree of differences supported by the data

Better practice: Use the CI to communicate the degree of uncertainty.


Hypothesis Testing, formalized

• DEFINITIONS:

▫ Hypothesis: A statement about a population parameter. Example: mean

bilirubin.

▫ Null Hypothesis (H0): The default claim.

Example: mean bilirubin = 3.

▫ Alternative Hypothesis (H1): The competing claim.

Example: mean bilirubin > 3 (one-sided)

Example: mean bilirubin < 3 (one-sided)

Example: mean bilirubin ≠ 3 (double-sided or two-sided)

Hypothesis Testing

• DEFINITIONS:


bilirubin.







Hypothesis Testing

• DEFINITIONS:


bilirubin.







Only interested in detecting differences in one direction

mean bilirubin > 2 (one-sided)

Ignored Only care about differences greater than 2

mean bilirubin < 2 (one-sided)

Ignored Only care about differences less than 2

mean bilirubin ≠ 2 (two-sided)The usual setup

Care to detect mean differences in any direction.

In general, for two-sided test with point null H0: mean = 2 H1: mean ≠ 2

Conclusive difference

You will see the phrase “rejected the null” when a conclusive difference is identified.

In general, for two-sided test with point null H0: mean = 2 H1: mean ≠ 2

Conclusive difference

You will see the phrase “rejected the null” when a conclusive difference is identified.

In my opinion, it is more straightforward to write: “detected a difference in …”

Hypothesis Testing

ONE-SIDED OR TWO-SIDED HYPOTHESIS TESTS

Remember, if there is a point null: One cannot identify conclusive

similarities. Therefore, one does not “accept the null” nor

“declare it to be true”.

One can perform an hypothesis test using a confidence interval.

How to perform an hypothesis test with point null via CI

• Compute a 1-α confidence interval for the parameter of

interest

Is the null hypothesis in the

CI?

Yes

No

Fail to reject H0

Reject H0

(Inconclusive result)

(Conclusive difference)

We saw this earlier … now updated with hypothesis testing vocab.

Reject H0 (Conclusive difference)Fail to reject H0 (Inconclusive result)

Hypothesis Testing via CI

Example: Low Birthweight Study

Suppose there is a concern that birth weights of infants among

mothers who smoked during pregnancy may be below the

national average. Suppose the national average is 3100g, and

we will consider it a population mean.

Hypothesis Testing via CI

Example (cont):

To assess our study question, we will express our test hypotheses as:

Among smoking mothers,

H0: mean birthweight = 3100g

H1: mean birthweight ≠ 3100g.

Hypothesis Testing vi CI

Example (side note):

Why is H1 ≠ 3100g instead of < 3100g? Even though our concern is low

birthweight infants, if smoking mothers are having babies that are

much heavier than the national average, there may be some other

unrealized problem.

Hypothesis testing via CI

bwt 74 2773.243 76.73218 2620.316 2926.17 Variable Obs Mean Std. Err. [95% Conf. Interval]

. ci means bwt if smoke == 1

What does one conclude from the CI?

Example (cont):

Calculate the 95% CI:

How to perform an hypothesis test via CI

Example (cont):

Is the null hyp (3100g) in the CI?

Yes

No

Fail to reject H0

Reject H095% CI: (2620, 2926)

Could our inference be wrong?(Yes)

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