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Shrikant R. Bharadwaj

BSopt., PhD

HYPOTHESIS TESTING
AND P – VALUE
Hypothesis Testing & p-value
This presentation will cover …
•

Hypothesis testing

•

Attributes of a sampling distribution

•

p-value

•

Type-I and Type-II errors in hypothesis testing
What is a Hypothesis?
•

Hypothesis is a proposed explanation of a phenomenon that can be
scientifically tested

•

Hypothesis is a tentative statement about the relationship between two
or more variables that is specific and testable

•

Evolution Vs. Creation controversy

•

Organisms evolve from one form to another is a testable hypothesis
proposed by Sir Charles Darwin

•

Organisms were created by a supernatural force (aka God) is a belief
and not a testable hypothesis

•

Tammy Kitzmiller, et al Vs. Dover Area Public School, et al (2005)

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What is a Hypothesis?
•

Hypothesis is a proposed explanation of a phenomenon that can be
scientifically tested

•

Hypothesis is a tentative statement about the relationship between two
or more variables that is specific and testable

•

Drugs that lower IOP reduce retinal ganglion cell loss

•

Using 3 doses of Avastin injection reduces retinal angiogenesis by 50%

•

The number of people entering Patodia hall for morning class is
maximum between 6:59:50AM and 7:00:00AM

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Null Vs. Alternate Hypothesis
•

Science is all about testing a given hypothesis

•

Two contradictory hypotheses under consideration
- Null Hypothesis (H0)
- Alternate Hypothesis (Ha)

•

Null hypothesis is typically the claim that is initially assumed to the true
- It is the default choice

•

Alternate hypothesis is typically opposite of the Null hypothesis

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Examples of Null & Alternate Hypothesis
•

One is considered innocent unless proven guilty
- Null hypothesis (H0): A person accused of murder is innocent
- Alternate hypothesis (Ha): This person is guilty of murder

•

What is the impact of an IOP lowering drug on retinal ganglion cell
loss?
- H0: Drug lowering IOP has no impact on retinal ganglion cell loss
- Ha: Drug lowering IOP has some impact on retinal ganglion cell
loss

•

What is the impact of Avastin on retinal angiogenesis?
- H0: Avastin has no impact on angiogensis
- Ha: Avastin has some impact on angiogensis

•

The alternate hypothesis is typically bi-directional (aka two-tailed)

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Null Vs. Alternate Hypothesis
What is the impact of beta
blockers on IOP?
Null hypothesis (H0)
The IOP in a placebo and
beta-blocker treated cohort
are not different from each
other
Alternate hypothesis (Ha)
The IOP in the beta-blocker
treated cohort is different from
the IOP in the placebo cohort
Mean of treatment group is lower than the placebo group
Lower-tail of the Placebo cohort’s Gaussian distribution
Null Vs. Alternate Hypothesis
•

The purpose of a study is to provide evidence for or against the null
hypothesis

•

Based on the evidence gathered by the study, you either support or reject
the null hypothesis

•

Only as a corollary, you reject or support the alternate hypothesis

Terminology clarification
•

You cannot PROVE the null hypothesis; you can only DISPROVE it

•

Science and hypothesis testing are based on the logic of falsification

•

http://www.statisticalmisconceptions.com/sample2.html

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Proving Vs. Disproving
•

Null hypothesis: All crows in this world are black

•

To PROVE the null hypothesis, you need to get the color of every single
crow in this world

•

To DISPROVE the null hypothesis, you just need to show one white
crow
Proving Vs. Providing Evidence
•

PROVE is a dangerous word – it leaves no room for error!!

•

(a + b)2 = a2 + b2 + 2ab --------- this can be PROVED mathematically

•

What is the impact of beta
blockers on IOP?

•

You are NOT PROVING that beta
blockers reduce IOP

•

You are only PROVIDING
EVIDENCE that beta blockers can
reduce IOP

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Proving Vs. Providing Evidence
Reasons why biological research
cannot PROVE anything
1.Humans react differently to a given
treatment
2.Measurement error
3.Data is not obtained from every
human being on Earth
Biological research can only
determine how likely or unlikely a
given result is

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Sampling a population
•

Data cannot be obtained from every human being on Earth

•

A representative cohort is sampled and results from this cohort are
extrapolated to the entire population

Fully deterministic distribution
with no standard deviation

Realistic biological distribution
with standard deviation
Properties of a Sampling Distribution

μ = Mean of Gaussian distribution; σ = Standard deviation of Gaussian distribution
Data from 68.2% of the population falls within +/-1σ
Data from 95.4% of the population falls within +/-2σ
Data from 99.6% of the population falls within +/-3σ
Standard Deviation & Confidence Intervals

Standard deviation describes variability of measurements in your sample
Confidence intervals describe the interval over which the mean will fall when the
experiment is repeated multiple times
95% Confidence interval = +/-1.96 SD
99% Confidence interval = +/-2.58 SD
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Z-scores

Z-score is a unitless quantity that describes how many standard deviations
away from the mean is your sample value
Z = (x – μ) / σ
1Z-score = 1SD; 2Z-scores = 2SD; 3Z-scores = 3SD
p-value
Biological research aims at determining the likelihood of the null hypothesis
being rejected
What is the likelihood that a lowered IOP was really due to the treatment and
not by chance?
p-value (or “probability” value) gives us this likelihood
p-value ranges from 0 to 1 or 0% to 100%
There can be 0% probability to 100% probability of rejecting the null
hypothesis

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p-value

p-value estimates the false positive
rate (Type 1 error) that we are willing
to accept
Typically, we accept a false-positive
rate of <=5% (p <= 0.05)
95% confidence that the IOP value
came from the treated distribution
95% confidence that null hypothesis
can be rejected
5% (or 1 in 20 times) our results can
be incorrect
p-value

p = 0.01

•99% confidence that null hypothesis
can be rejected
•1% (or 1 in 100 times) our results
can be incorrect
p = 0.1

•90% confidence that null hypothesis
can be rejected
•10% (or 1 in 10 times) our results
can be incorrect
Determinants of the p-value
p-value is lower when…
1.Mean difference is large
2.Small variance in each distribution
Example 1:
Non diseased Mean + 1SD: 20 + 5mmHg
Diseased Mean + 1SD: 25 + 5mmHg
Example 2:
Non diseased Mean + 1SD: 20 + 5mmHg
Diseased Mean + 1SD: 35 + 5mmHg
Example 3:
Non diseased Mean + 1SD: 20 + 2mmHg
Diseased Mean + 1SD: 35 + 2mmHg
p-value of Eg 3 < Eg 2 < Eg 1
Determinants of the p-value
p-value is lower when…
1.Mean difference is large
2.Small variance in each distribution
Example 1:
Non diseased Mean + 1SD: 20 + 5mmHg
Diseased Mean + 1SD: 25 + 5mmHg
Example 2:
Non diseased Mean + 1SD: 20 + 5mmHg
Diseased Mean + 1SD: 35 + 5mmHg
Example 3:
Non diseased Mean + 1SD: 20 + 2mmHg
Diseased Mean + 1SD: 35 + 2mmHg
p-value of Eg 3 < Eg 2 < Eg 1
Use of p-value in a Student’s t-test

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Use of p-value in a Student’s t-test
•

Two kinds of t-tests

•

Paired t-test: The two datasets are obtained from the same cohort (e.g.
IOP before and after treatment with beta-blockers)

•

Unpaired t-test: The two datasets are obtained on different cohorts (e.g.
Body weight of 30 – 40yr old males and females)

Practical demo of T-test in MS Excel
Large mean
difference

Small mean
difference

Large variance

p = 0.0325

p = 0.2906

Small variance

p < 0.0001

p = 0.1011
Type-I and Type-II Errors
•

Type-I error (α error): When the Null hypothesis is true, but it is rejected by
the test
• Type-I error is equivalent to generating False Positives

•

Type-II error (β error): When the Null hypothesis is false, but it is
erroneously accepted as true
• Type-II error is equivalent to generating False Negatives

Null hypothesis is true
Reject Null
hypothesis

Null hypothesis is false

Type-I error / FP

Correct decision / TP

ReferAccept Nullin my first presentation for/ its equivalent in diagnostic tests
to slide #8
Correct rejection TN
Type-II error / FN
hypothesis

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Example of Type-I and Type-II Errors
The radio engineer during WW II receives a crackling sound over his
transmitter. Is this signal from the enemy or is it unwanted noise?
Null hypothesis: The sound received over the transmitter is just noise
Alternate hypothesis: The sound received over the transmitter is not noise
True Signal

Just Noise

Interpretation
as Signal

Correctly rejecting null
hypothesis

False rejecting null
hypothesis / Type-I error

Interpretation
as Noise

Falsely accepting null
hypothesis / Type-II
error

Correctly accepting the
null hypothesis
Example for Type-II error
What is the difference in macular thickness of eyes with AMD compared
to normals, as detected using OCT imaging?
•Null hypothesis: There is no difference in macular thickness between normal
eyes and eyes with AMD
•Alternate hypothesis: The macula in AMD patients is >50μ thicker than it is in
normal eyes
•Mean + 1SD macular thickness in normal eyes: 200μ + 50μ
•Mean + 1SD macular thickness in AMD eyes: 230μ + 55μ
•Based on these results, you have accepted the null hypothesis

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Example for Type-II error
•

Repeatability of the OCT: 80μ

•

The test that you have used does not have the resolution to determine the
difference you are expecting

•

The mean difference in macular thickness between normal and AMD eyes
is 110μ using a gold-standard test

•

An Type-II error is therefore made, in erroneously accepting the Null
hypothesis

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Thank You!

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Hypothesis testing and p-value, www.eyenirvaan.com

  • 1. Shrikant R. Bharadwaj BSopt., PhD HYPOTHESIS TESTING AND P – VALUE
  • 3. This presentation will cover … • Hypothesis testing • Attributes of a sampling distribution • p-value • Type-I and Type-II errors in hypothesis testing
  • 4. What is a Hypothesis? • Hypothesis is a proposed explanation of a phenomenon that can be scientifically tested • Hypothesis is a tentative statement about the relationship between two or more variables that is specific and testable • Evolution Vs. Creation controversy • Organisms evolve from one form to another is a testable hypothesis proposed by Sir Charles Darwin • Organisms were created by a supernatural force (aka God) is a belief and not a testable hypothesis • Tammy Kitzmiller, et al Vs. Dover Area Public School, et al (2005) To view more presentations and articles, visit www.eyenirvaan.com
  • 5. What is a Hypothesis? • Hypothesis is a proposed explanation of a phenomenon that can be scientifically tested • Hypothesis is a tentative statement about the relationship between two or more variables that is specific and testable • Drugs that lower IOP reduce retinal ganglion cell loss • Using 3 doses of Avastin injection reduces retinal angiogenesis by 50% • The number of people entering Patodia hall for morning class is maximum between 6:59:50AM and 7:00:00AM To view more presentations and articles, visit www.eyenirvaan.com
  • 6. Null Vs. Alternate Hypothesis • Science is all about testing a given hypothesis • Two contradictory hypotheses under consideration - Null Hypothesis (H0) - Alternate Hypothesis (Ha) • Null hypothesis is typically the claim that is initially assumed to the true - It is the default choice • Alternate hypothesis is typically opposite of the Null hypothesis To view more presentations and articles, visit www.eyenirvaan.com
  • 7. Examples of Null & Alternate Hypothesis • One is considered innocent unless proven guilty - Null hypothesis (H0): A person accused of murder is innocent - Alternate hypothesis (Ha): This person is guilty of murder • What is the impact of an IOP lowering drug on retinal ganglion cell loss? - H0: Drug lowering IOP has no impact on retinal ganglion cell loss - Ha: Drug lowering IOP has some impact on retinal ganglion cell loss • What is the impact of Avastin on retinal angiogenesis? - H0: Avastin has no impact on angiogensis - Ha: Avastin has some impact on angiogensis • The alternate hypothesis is typically bi-directional (aka two-tailed) To view more presentations and articles, visit www.eyenirvaan.com
  • 8. Null Vs. Alternate Hypothesis What is the impact of beta blockers on IOP? Null hypothesis (H0) The IOP in a placebo and beta-blocker treated cohort are not different from each other Alternate hypothesis (Ha) The IOP in the beta-blocker treated cohort is different from the IOP in the placebo cohort Mean of treatment group is lower than the placebo group Lower-tail of the Placebo cohort’s Gaussian distribution
  • 9. Null Vs. Alternate Hypothesis • The purpose of a study is to provide evidence for or against the null hypothesis • Based on the evidence gathered by the study, you either support or reject the null hypothesis • Only as a corollary, you reject or support the alternate hypothesis Terminology clarification • You cannot PROVE the null hypothesis; you can only DISPROVE it • Science and hypothesis testing are based on the logic of falsification • http://www.statisticalmisconceptions.com/sample2.html To view more presentations and articles, visit www.eyenirvaan.com
  • 10. Proving Vs. Disproving • Null hypothesis: All crows in this world are black • To PROVE the null hypothesis, you need to get the color of every single crow in this world • To DISPROVE the null hypothesis, you just need to show one white crow
  • 11. Proving Vs. Providing Evidence • PROVE is a dangerous word – it leaves no room for error!! • (a + b)2 = a2 + b2 + 2ab --------- this can be PROVED mathematically • What is the impact of beta blockers on IOP? • You are NOT PROVING that beta blockers reduce IOP • You are only PROVIDING EVIDENCE that beta blockers can reduce IOP To view more presentations and articles, visit www.eyenirvaan.com
  • 12. Proving Vs. Providing Evidence Reasons why biological research cannot PROVE anything 1.Humans react differently to a given treatment 2.Measurement error 3.Data is not obtained from every human being on Earth Biological research can only determine how likely or unlikely a given result is To view more presentations and articles, visit www.eyenirvaan.com
  • 13. Sampling a population • Data cannot be obtained from every human being on Earth • A representative cohort is sampled and results from this cohort are extrapolated to the entire population Fully deterministic distribution with no standard deviation Realistic biological distribution with standard deviation
  • 14. Properties of a Sampling Distribution μ = Mean of Gaussian distribution; σ = Standard deviation of Gaussian distribution Data from 68.2% of the population falls within +/-1σ Data from 95.4% of the population falls within +/-2σ Data from 99.6% of the population falls within +/-3σ
  • 15. Standard Deviation & Confidence Intervals Standard deviation describes variability of measurements in your sample Confidence intervals describe the interval over which the mean will fall when the experiment is repeated multiple times 95% Confidence interval = +/-1.96 SD 99% Confidence interval = +/-2.58 SD To view more presentations and articles, visit www.eyenirvaan.com
  • 16. Z-scores Z-score is a unitless quantity that describes how many standard deviations away from the mean is your sample value Z = (x – μ) / σ 1Z-score = 1SD; 2Z-scores = 2SD; 3Z-scores = 3SD
  • 17. p-value Biological research aims at determining the likelihood of the null hypothesis being rejected What is the likelihood that a lowered IOP was really due to the treatment and not by chance? p-value (or “probability” value) gives us this likelihood p-value ranges from 0 to 1 or 0% to 100% There can be 0% probability to 100% probability of rejecting the null hypothesis To view more presentations and articles, visit www.eyenirvaan.com
  • 18. p-value p-value estimates the false positive rate (Type 1 error) that we are willing to accept Typically, we accept a false-positive rate of <=5% (p <= 0.05) 95% confidence that the IOP value came from the treated distribution 95% confidence that null hypothesis can be rejected 5% (or 1 in 20 times) our results can be incorrect
  • 19. p-value p = 0.01 •99% confidence that null hypothesis can be rejected •1% (or 1 in 100 times) our results can be incorrect p = 0.1 •90% confidence that null hypothesis can be rejected •10% (or 1 in 10 times) our results can be incorrect
  • 20. Determinants of the p-value p-value is lower when… 1.Mean difference is large 2.Small variance in each distribution Example 1: Non diseased Mean + 1SD: 20 + 5mmHg Diseased Mean + 1SD: 25 + 5mmHg Example 2: Non diseased Mean + 1SD: 20 + 5mmHg Diseased Mean + 1SD: 35 + 5mmHg Example 3: Non diseased Mean + 1SD: 20 + 2mmHg Diseased Mean + 1SD: 35 + 2mmHg p-value of Eg 3 < Eg 2 < Eg 1
  • 21. Determinants of the p-value p-value is lower when… 1.Mean difference is large 2.Small variance in each distribution Example 1: Non diseased Mean + 1SD: 20 + 5mmHg Diseased Mean + 1SD: 25 + 5mmHg Example 2: Non diseased Mean + 1SD: 20 + 5mmHg Diseased Mean + 1SD: 35 + 5mmHg Example 3: Non diseased Mean + 1SD: 20 + 2mmHg Diseased Mean + 1SD: 35 + 2mmHg p-value of Eg 3 < Eg 2 < Eg 1
  • 22. Use of p-value in a Student’s t-test To view more presentations and articles, visit www.eyenirvaan.com
  • 23. Use of p-value in a Student’s t-test • Two kinds of t-tests • Paired t-test: The two datasets are obtained from the same cohort (e.g. IOP before and after treatment with beta-blockers) • Unpaired t-test: The two datasets are obtained on different cohorts (e.g. Body weight of 30 – 40yr old males and females) Practical demo of T-test in MS Excel Large mean difference Small mean difference Large variance p = 0.0325 p = 0.2906 Small variance p < 0.0001 p = 0.1011
  • 24. Type-I and Type-II Errors • Type-I error (α error): When the Null hypothesis is true, but it is rejected by the test • Type-I error is equivalent to generating False Positives • Type-II error (β error): When the Null hypothesis is false, but it is erroneously accepted as true • Type-II error is equivalent to generating False Negatives Null hypothesis is true Reject Null hypothesis Null hypothesis is false Type-I error / FP Correct decision / TP ReferAccept Nullin my first presentation for/ its equivalent in diagnostic tests to slide #8 Correct rejection TN Type-II error / FN hypothesis To view more presentations and articles, visit www.eyenirvaan.com
  • 25. Example of Type-I and Type-II Errors The radio engineer during WW II receives a crackling sound over his transmitter. Is this signal from the enemy or is it unwanted noise? Null hypothesis: The sound received over the transmitter is just noise Alternate hypothesis: The sound received over the transmitter is not noise True Signal Just Noise Interpretation as Signal Correctly rejecting null hypothesis False rejecting null hypothesis / Type-I error Interpretation as Noise Falsely accepting null hypothesis / Type-II error Correctly accepting the null hypothesis
  • 26. Example for Type-II error What is the difference in macular thickness of eyes with AMD compared to normals, as detected using OCT imaging? •Null hypothesis: There is no difference in macular thickness between normal eyes and eyes with AMD •Alternate hypothesis: The macula in AMD patients is >50μ thicker than it is in normal eyes •Mean + 1SD macular thickness in normal eyes: 200μ + 50μ •Mean + 1SD macular thickness in AMD eyes: 230μ + 55μ •Based on these results, you have accepted the null hypothesis To view more presentations and articles, visit www.eyenirvaan.com
  • 27. Example for Type-II error • Repeatability of the OCT: 80μ • The test that you have used does not have the resolution to determine the difference you are expecting • The mean difference in macular thickness between normal and AMD eyes is 110μ using a gold-standard test • An Type-II error is therefore made, in erroneously accepting the Null hypothesis To view more presentations and articles, visit www.eyenirvaan.com
  • 28. Thank You! To view more presentations and articles, visit www.eyenirvaan.com