P value, Power, Type 1 and 2 errors

P value, Power & Type I & II error
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 5 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh
• Define p value
• Describe the meaning and limitations of p value
• Define power of a test and its meaning
• Describe type 1 and type 2 errors in hypothesis
testing and how they affect the interpretation of
results
• Understand how consideration of p value, type 1
and 2 errors relate to sample size calculation
2

Section 1:
P value
Demystifying statistics! – Lecture 5 SBCM, Joint Program – RiyadhSBCM, Joint Program – Riyadh 3

P value
• Defined as the probability of obtaining a result
equal to or more extreme than what was actually
observed
• First introduced by Karl Pearson in his Pearson's
chi-squared test
• It can also be seen in relation to the probability of
making a Type I error
4

P value
The vertical coordinate is the probability density of each outcome, computed under the null hypothesis.
The p-value is the area under the curve past the observed data point.
5

P value – choice of cut off value
• Arbitrary cut-off 0.05 (5% chance of a false+ conclusion)
• If p<0.05 statistically significant- Reject H0, Accept H1
• If p>0.05 statistically not significant, Accept H0, Reject H1
• Testing potential harmful interventions ‘α’ value is set
below 0.05
• Depends upon the research question!
6

P value – degrees of magnitude
• Very small (<0.001), the results are said to be
highly significant
• Near 0.05, it is said to be borderline significant
• Near 1.0, result does not matter!
7

P value – how to calculate it?
• Depending upon the statistic we are interested in
predetermined p values and their critical values
are displayed in statistical tables
• So each type of distribution has its own table
• It is also possible to calculate exact p values with
computers instead of using such tables
8

P value – interpretation
• If the results are statistically significant, decide whether the
observed differences are clinically important
• If not significant, see if the sample size was adequate enough
not to have missed a clinically important difference
• Power of the study tells us the strength which we can conclude
that there is no difference between the two groups
9

P value – interpretation
• Statistical significance does not necessarily mean real significance
• If sample size is large, even small differences can have a low p-value
• Lack of significance doesn’t necessarily mean null hypothesis is true
• If sample size is small, there could be a real difference, but we are not
able to detect it
• If you perform a large number of tests in a study, 1 in 20 will be
significant merely by chance
10

Section 2:
Type 1 and 2 errors

• These are errors that arise when performing hypothesis testing and
decision making
• Type 1 error (false positive conclusion)
• Stating difference when there is no difference, alpha
• Related to p value, how?
• Set at 1/20 or 0.05 or 5%
• The probability is distributed at the tails of the normal curve i.e., 0.025 on
either tail
• Type 2 error (false negative conclusion)
• Stating no difference when there is a difference, beta
• Occurs when sample size is too small.
• Conventional values are 0.1 or 0.2
• Related to power, how?
What are these errors?
12

What are these errors?
13
Reality:
No effect
Reality:
Effect exists
Research concludes:
Fail to reject null;
No effect
CORRECT FAILURE TO
REJECT
TYPE 2 ERROR (β)
Researcher concludes:
Reject null;
Effect exists
TYPE 1 ERROR (α) CORRECT REJECT (1-β)
• Advanced learning: Do you know there are type 3 and 4 also?

Example 1
14

Example 2
15

Example 3
16

Section 3:
Power of the study

Power of the study
• The ability to detect a statistically significant association
• It can also be seen as the probability of not missing an effect,
due to sampling error, when there really is an effect
• It is also the probability of avoiding a type 2 error, i.e., 1 – beta
• A prospective power analysis is used before collecting data, to
consider design sensitivity
• A retrospective power analysis is used in order to know
whether the studies you are interpreting were well enough
designed
18

Factors affecting power
• All else being equal:
1. As sample sizes increase, power increases
2. As population variances decrease, power increases
3. As the difference increases, power increases
4. Statistical power is greater for one-tailed tests
5. The greater the probability of making a Type I error, the
greater the power
19

Calculating Power: Example
• A study of n = 16 retains null H: μ = 170 at α = 0.05 (two-sided);
σ is 40. What was the power of test’s conditions to identify a
population mean of 190?
( )
5160.0
04.0
40
16|190170|
96.1
||
1 0
1 2
=
Φ=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
+−Φ=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
+−Φ=− −
σ
µµ
β α
n
z a
20

Calculating Power: Example
• Top curve assumes null H is true
• Bottom curve assumes alternative H is true
• α is set to 0.05 (two-sided)
• We will reject null when a sample mean exceeds
189.6 (right tail, top curve)
• The probability of getting a value greater than 189.6
on the bottom curve is 0.5160, corresponding to the
power of the test
21

Power vs. confidence intervals
• Once we have constructed a confidence interval, power calculations
yield no additional insights
• It is pointless to perform power calculations for hypotheses outside
of the confidence interval
• Confidence intervals better inform readers about the possibility of
an inadequate sample size than do post hoc power calculations
22

How do the errors relate to sample size?
• Sample size for one-sample z test:
• 1 – β ≡ desired power
• α ≡ desired significance level (two-sided)
• σ ≡ population standard deviation
• Δ = μ0 – μa ≡ the difference worth detecting
( )
2
2
11
2
2
Δ
+
=
−− αβσ zz
n
23

• How large a sample is needed for a one-sample z test
with 90% power and α = 0.05 (two-tailed) when σ =
40? Let H0: μ = 170 and Ha: μ = 190 (thus, Δ = μ0 − μa
= 170 – 190 = −20)
• Sample size should be 42 to ensure adequate power.
( ) 99.41
20
)96.128.1(40
2
22
2
2
11
2
2
=
−
+
=
Δ
+
=
−− αβσ zz
n
24

N = 16 N = 42
25

Take home messages
• P value, type 1 and 2 errors,
alpha, beta, power, critical
value and hypothesis testing,
sample size are all related to
each other
26

Thank you!
Email your queries to sarizwan1986@outlook.com
27

P value, Power, Type 1 and 2 errors

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