Confidence Intervals: Basic concepts and overview

An brief introduction to confidence intervals in inferential statistics

1. Overview of Confidence Intervals
Dr.S. A. Rizwan, M.D.
Public Health Specialist
SBCM, Joint Program –Riyadh
Ministry of Health, Kingdom of Saudi Arabia

2. 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

3. 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

4. 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 pretestexam

5. 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 100samples from the same population

6. 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

7. Some important terms
Demystifying statistics! –Lecture 9 SBCM, Joint Program –RiyadhSBCM, Joint Program –Riyadh
•Point estimate
•Standard error
•Confidence level

8. Revise: standard deviation
Demystifying statistics! –Lecture 9 SBCM, Joint Program –RiyadhSBCM, Joint Program –Riyadh
€
ˆ σ =s=
(Y
i
−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

9. 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

10. Revise: standard error
Demystifying statistics! –Lecture 9 SBCM, Joint Program –RiyadhSBCM, Joint Program –Riyadh

11. 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)

12. 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

13. What is CI?
Demystifying statistics! –Lecture 9 SBCM, Joint Program –RiyadhSBCM, Joint Program –Riyadh

14. 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
•9% -2.58
Point Estimate ±(Critical Value) * (Standard Error)

15. 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

16. 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”

17. 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

18. 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

19. 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

20. 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)

21. 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.

22. 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 newbornbirth weights are between 3181 and 3319g
b)The mean birth weight for these 153 newbornsis probably between 3181 and 3319g
c)The mean of the population from which the 153 newbornscame is between 3181 and 3319g
d)None of the above

23. Various interpretations of CI
Demystifying statistics! –Lecture 9 SBCM, Joint Program –RiyadhSBCM, Joint Program –Riyadh
•The confidence level does NOTtell us the chance that a
particular confidence interval captures the population
parameter.
•We CANNOTassign 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

24. Various interpretations of CI
Demystifying statistics! –Lecture 9 SBCM, Joint Program –RiyadhSBCM, Joint Program –Riyadh
•We CANsay:
•We are 95% confident that the confidence interval
calculated from our sample will contain the
population value
•We CANNOTsay:
•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

25. 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

26. 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

27. 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

28. 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%

29. 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

30. 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

31. Factors affecting CI
Demystifying statistics! –Lecture 9 SBCM, Joint Program –RiyadhSBCM, Joint Program –Riyadh

32. 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.

33. 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.

34. Assumptions for CI
Demystifying statistics! –Lecture 9 SBCM, Joint Program –RiyadhSBCM, Joint Program –Riyadh
•Independent:
•Individual observations are independent

35. 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

36. 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

37. 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

38. How does CI relate to sample size?
Demystifying statistics! –Lecture 9 SBCM, Joint Program –RiyadhSBCM, Joint Program –Riyadh

39. 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

40. How does CI relate to significance level?
Demystifying statistics! –Lecture 9 SBCM, Joint Program –RiyadhSBCM, Joint Program –Riyadh

41. 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

42. Thank you!
Email your queries to [email protected]