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Confidence Intervals: Basic concepts and overview

This document provides an overview of confidence intervals. It defines confidence intervals and describes their use in statistical inference to estimate population parameters. It explains that a confidence interval provides a range of plausible values for an unknown population parameter based on a sample statistic. The document outlines the key steps in calculating a confidence interval, including determining the point estimate, standard error, and critical value corresponding to the desired confidence level. It discusses how the width of the confidence interval indicates the precision of the estimate and is affected by factors like the sample size and population variability.

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Overview of Confidence Intervals Dr. S. A. Rizwan, M.D. Public Health Specialist SBCM, Joint Program – Riyadh Ministry of Health, Kingdom of Saudi Arabia
Learning objectives Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Define confidence intervals • Describe their use in statistical inference • Describe and apply the steps in calculating CI
Statistical inference Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Statistical inference - drawing conclusions about a population from sample • Methods • Confidence Intervals - estimating a value of a population parameter • Tests of significance - assess evidence for a claim about a population
Thought exercises Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Estimation of a population mean • Mean score obtained by this class in the pretest exam
Thought exercises Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 95 of these 100 CIs will contain the population parameter There are 100 sample means and 100 CIs Calculate sample statistic eg. mean for each sample Take 100 samples from the same population
Thought exercises Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • We don’t need to take a lot of random samples to “rebuild” the sampling distribution • All we need is one SRS of size n and rely on the properties of the sample means distribution to infer the population mean
Some important terms Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Point estimate • Standard error • Confidence level
Revise: standard deviation Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh € ˆσ = s = (Yi −Y )2 ∑ n −1 • How much your data is spread out around average • For example, are all your scores close to the average? Or are lots of scores way above (or way below) the average score? For Means For proportions
Revise: standard error Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • This is not the standard deviation of the sample, it is the standard deviation of the sample distribution of proportions (or means) For Means For proportions
Revise: standard error Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
Why CI? Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • A point estimate provides no information about the precision and reliability of estimation • A point estimate says nothing about how close it might be to μ • An alternative to reporting a single sensible value is to calculate and report an entire interval of plausible values – a confidence interval (CI)
What is CI? Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • An interval gives a range of values: • Takes into consideration variation in sample statistics from sample to sample • Based on observations from 1 sample • Gives information about closeness to unknown population parameters • Stated in terms of level of confidence. • Can never be 100% confident • An interval of values computed from the sample, that is almost sure to cover the true population value
What is CI? Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
General format of CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Z values for different Confidence levels • 90% - 1.64 • 95% - 1.96 • 98% - 2.33 • 99% - 2.58 Point Estimate ± (Critical Value) * (Standard Error)
Various interpretations of CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • In 95% of the samples we take, the true population proportion (or mean) will be in the interval • We are 95% confident that the true population proportion (or mean) will be in the interval • In 95% of all possible samples of this size n, µ will indeed fall in our confidence interval
Various interpretations of CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • In only 5% of samples would sample mean be farther from µ • To say that we are 95% confident is shorthand for “95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter.” • To interpret a C% confidence interval for an unknown parameter, say, “We are C% confident that the interval from _____ to _____ captures the actual value of the population parameter”
Various interpretations of CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • A confidence interval provides additional information about variability • For a 95% confidence interval about 95% of the similarly constructed intervals will contain the parameter being estimated. • Also 95% of the sample means for a specified sample size will lie within 1.96 standard deviations of the hypothesized population
Various interpretations of CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • In general, we construct such intervals so that, should we repeat the process a large number of times, then 95%, for a 95% confidence interval, of such intervals should contain the population parameter being estimated by the point estimate and the confidence interval
Various interpretations of CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • The specific interval we compute in any given situation may or may not contain the population parameter • The only way for us to be sure that the population parameter is within the bounds of the confidence interval is to know the true value for this parameter • Obviously, if we knew the true value, we would not bother to go through the process of guessing at the truth with estimates
Various interpretations of CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Example: 0.05 (0.036, 0.064) • Correct: • We are 95% confident that the interval from 0.036 to 0.064 actually does contain the true value • This means that if we were to select many different samples of size 1000 and construct a 95% CI from each sample, 95% of the resulting intervals would contain the population value • (0.036, 0.064) is one such interval. (Note that 95% refers to the procedure we used to construct the interval; it does not refer to the population value) • Wrong: There is a 95% chance that the population value falls between 0.036 and 0.064. (Note that p is not random, it is a fixed but unknown number)
Various interpretations of CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • You have measured the systolic blood pressure of a random sample of 30 employees of a company. A 95% confidence interval for the mean systolic blood pressure for the employees is computed to be (122, 138). Which of the following statements gives a valid interpretation of this interval? a) 95% of the sample of employees has a systolic blood pressure between 122 and 138. b) 95 % of the employees in the company have a systolic blood pressure between 122 and 138. c) If the sampling procedure were repeated 100 times, then approximately 95 of the sample means would be between 122 and 138. d) If the sampling procedure were repeated 100 times, then approximately 95 of the resulting 100 confidence intervals would contain the true mean systolic blood pressure for all employees of the company. e) We are 95% confident the sample mean is between 122 and 138.
Various interpretations of CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • The mean and standard deviation of the birth weights of a representative sample of 153 newborns are 3250 grams and 428 grams respectively. On the basis of these figures, a 95% confidence interval for the population mean birth weight runs from 3181 to 3319 grams. a) About 95% of the individual newborn birth weights are between 3181 and 3319g b) The mean birth weight for these 153 newborns is probably between 3181 and 3319g c) The mean of the population from which the 153 newborns came is between 3181 and 3319g d) None of the above
Various interpretations of CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • The confidence level does NOT tell us the chance that a particular confidence interval captures the population parameter. • We CANNOT assign probability to the population value because it is fixed and does not change depending on our sample values. • Width of the interval – indicates variability in the data
Various interpretations of CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • We CAN say: • We are 95% confident that the confidence interval calculated from our sample will contain the population value • We CANNOT say: • There is a 95% probability or chance that the confidence interval will contain the population value • There is a 95% probability or chance the population value will lie in this confidence interval • 95% of the time the population value will lie in this confidence interval
Interpretation of CI in comparative situations Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Null value within the limits of the CI • 0 for differences and 1 for ratios
Interpretation of CI in comparative situations Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • The mother who smoke had significantly higher risk (RR= 2.1; 1.8, 2.6, p=0.01) of having LBW babies and compared to those who did not smoke • Does the interval contain null value= No; association is significant • Width of the interval- variability in the estimate was less
Interpretation of CI in comparative situations Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • The mother who smoke had significantly higher risk (RR= 2.1, 0.8, 4.9, p=0.06) of having LBW babies and compared to those who did not smoke • Does the interval contain null value= Yes; association is insignificant • Width of the interval= high variability in the sample estimate
Thought exercise Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Series of 5 trials • Equal duration • Different sample sizes • To determine whether a novel drug is better than placebo in preventing stroke • Smallest trial has 8 patients • Largest trial has 2000 patients • Half of the patients in each trial – New drug • All trials - Relative risk reduction by 50%
Thought exercise Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Questions: • In each individual trial, how confident can we be regarding the relative risk reduction? • Larger trials - more confident • Which trials would lead you to recommend the treatment unequivocally to your patients? • CI - Range within which the true effect of test drug might plausibly lie in the given trial data
Factors affecting CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Factors that determine the width of a confidence interval are: • Sample size, n • Variability in the population • Desired level of confidence • The higher the confidence level, the more strongly we believe that the true value of the parameter being estimated lies within the interval
Factors affecting CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
Assumptions for CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Random: The data should come from a well-designed random sample or randomized experiment.
Assumptions for CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Normal: The sampling distribution of the statistic is approximately Normal. • For means: • The sampling distribution is exactly Normal if the population distribution is Normal. • When the population distribution is not Normal, then the central limit theorem tells us the sampling distribution will be approximately Normal if n is sufficiently large (n ≥ 30). • For proportions: • We can use the Normal approximation to the sampling distribution as long as np ≥ 10 and n(1 – p) ≥ 10.
Assumptions for CI Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Independent: • Individual observations are independent
How does CI relate to sample size? Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • Cost is directly proportional to sample size, so we generally want the minimum sample to do the job • Estimating minimum sample size is commonly done with population proportions • With population proportions, you do not need to make separate guesses about the population mean and standard deviation • With population proportions, it is easy to identify a conservative mean, and the bias does not vary much
How does CI relate to sample size? Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • For mean • When we choose the best sample size, we choose one half of the confidence interval (the top one) and solve for n n s zYic ±=.. 2 2/1 2 2 )..( µ σ − = topic zn
How does CI relate to sample size? Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • For proportion • When we choose the best sample size, we choose one half of the confidence interval (the top one) and solve for n n zic )ˆ1(ˆ ˆ.. ππ π − ±= 2 2/1 2 )..( )1( π ππ − − = topic zn
How does CI relate to sample size? Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
How does CI relate to significance level? Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh Confidence Level ‘z’ Value ‘a’ / 2 Value 80% 1.28 .1000 90% 1.64 .0500 95% 1.96 .0250 98% 2.33 .0100 99% 2.58 .0050 99.8% 3.08 .0010 99.9% 3.27 .0005
How does CI relate to significance level? Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
Take home messages Demystifying statistics! – Lecture 9 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh • P value, critical value, alfa, type 1 error, confidence interval, sample size are all related to each other
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