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Basic Statistical Concepts

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Chapter 2 Reading instructions

- 2.1 Introduction: Not very important
- 2.2 Uncertainty and probability: Read
- 2.3 Bias and variability: Read
- 2.4 Confounding and interaction: Read
- 2.5 Descriptive and inferential statistics: Repetition
- 2.6 Hypothesis testing and p-values: Read
- 2.7 Clinical significance and clinical equivalence: Read
- 2.8 Reproducibility and generalizability: Read

Bias and variability

Bias: Systemtic deviation from the true value

Design, Conduct, Analysis, Evaluation

Lots of examples on page 49-51

-10

-10

-7

-7

-7

-4

-4

-4

Bias and variabilityLarger study does not decrease bias

Population mean

;

bias

Distribution of sample means:

= population mean

Drog X - Placebo

Drog X - Placebo

Drog X - Placebo

mm Hg

mm Hg

mm Hg

n=200

N=2000

n=40

Bias and variability

There is a multitude of sources for bias

Publication bias

Positive results tend to be published while negative of inconclusive results tend to not to be published

Selection bias

The outcome is correlated with the exposure. As an example, treatments tends to be prescribed to those thought to benefit from them. Can be controlled by randomization

Exposure bias

Differences in exposure e.g. compliance to treatment could be associated with the outcome, e.g. patents with side effects stops taking their treatment

Detection bias

The outcome is observed with different intensity depending no the exposure. Can be controlled by blinding investigators and patients

Analysis bias

Essentially the I error, but also bias caused by model miss specifications and choice of estimation technique

Interpretation bias

Strong preconceived views can influence how analysis results are interpreted.

Bias and variability

Amount of difference between observations

True biological:

Variation between subject due to biological factors (covariates) including the treatment.

Temporal:

Variation over time (and space)

Often within subjects.

Measurement error:

Related to instruments or observers

Design, Conduct, Analysis, Evaluation

Confounding

Confounders

Predictors of outcome

Predictors of treatment

A

Treatment

allocation

Outcome

B

Example

Smoking Cigaretts is not so bad but watch out for Cigars or Pipes (at least in Canada)

Cochran, Biometrics 1968

*) per 1000 person-years %

Example

Smoking Cigaretts is not so bad but watch out for Cigars or Pipes (at least in Canada)

*) per 1000 person-years %

Cochran, Biometrics 1968

Example

Smoking Cigaretts is not so bad but watch out for Cigars or Pipes (at least in Canada)

*) per 1000 person-years %

Cochran, Biometrics 1968

Confounding

The effect of two or more factors can not be separated

Example:

Compare survival for surgery and drug

Survival

Life long treatment with drug

R

Surgery at time 0

Time

- Surgery only if healty enough
- Patients in the surgery arm may take drug
- Complience in the drug arm May be poor

Looks ok but:

R

M

A

B

R

Gender

A

B

R

F

B

ConfoundingCan be sometimes be handled in the design

Example: Different effects in males and females

Imbalance between genders affects result

Stratify by gender

Balance on average

Always balance

B

Wash out

R

B

A

InteractionThe outcome on one variable depends on the value of another variable.

Example

Interaction between two drugs

A=AZD1234

B=AZD1234 + Clarithromycin

Interaction

Example: Drug interaction

AUC AZD0865:

19.75 (µmol*h/L)

AUC AZD0865 + Clarithromycin:

36.62 (µmol*h/L)

Ratio:

0.55 [0.51, 0.61]

Interaction

Example:

Treatment by center interaction

Average treatment effect: -4.39 [-6.3, -2.4] mmHg

Treatment by center: p=0.01

What can be said about the treatment effect?

Descriptive and inferential statistics

The presentation of the results from a clinical trial can be split in three categories:

- Descriptive statistics
- Inferential statistics
- Explorative statistics

Descriptive and inferential statistics

Descriptive statistics aims to describe various aspects of the data obtained in the study.

- Listings.
- Summary statistics (Mean, Standard Deviation…).
- Graphics.

Descriptive and inferential statistics

Inferential statistics forms a basis for a conclusion regarding a prespecified objective addressing the underlying population.

Confirmatory analysis:

Hypothesis

Results

Conclusion

Descriptive and inferential statistics

Explorative statistics aims to find interesting results that

Can be used to formulate new objectives/hypothesis for further investigation in future studies.

Explorative analysis:

Results

Hypothesis

Conclusion?

Hypothesis testing, p-values and confidence intervals

Objectives

Variable

Design

Estimate

p-value

Confidence interval

Statistical Model

Null hypothesis

Results

Interpretation

Statistical model: Observations

from a class of distribution functions

Hypothesis test: Set up a null hypothesis: H0:

and an alternative H1:

Hypothesis testing, p-valuesReject H0 if

Significance level

Rejection region

p-value:

The smallest significance level for which the null hypothesis can be rejected.

What is statistical hypothesis testing?

A statistical hypothesis test is a method of making decisions using data from a scientific study.

Decisions:

Conclude that substance is likely to be efficacious in a target population.

Scientific study:

Collect data measuring efficacy of a substance on a random sample of patients representative of the target population

Formalizing hypothesis testing

- We try to prove that collected data with little likelihood can correspond to a statement or a hypothesis which we do not want to be true!
- Hypothesis:
- The effect of our substance has no effect on blood pressure compared to placebo
- An active comparator is superior to our substance

- What do we want to be true then?
- The effect of our substance has a different effect on blood pressure compared to placebo
- Our substance is non-inferior to active comparator

Formalizing hypothesis testing cont.

Formalizing it even further…

As null hypothesis, H0, we shall choose the hypothesis we want to reject

H0: μX = μplacebo

As alternative hypothesis, H1, we shall choose the hypothesis we want to prove.

H1: μX ≠ μplacebo

(two sided alternative hypothesis)

μ is a parameter that characterize the efficacy in the target population, for example the Mean.

H0: Treatment with substance X has no effect on blood pressure compared to placebo

0

Difference in BP

H1: Treatment with substance X influence blood pressure compared to placebo

H0: Treatment with substance X has no effect on blood pressure compared to placebo

Difference in BP

0

H1: Treatment with substance X decrease the blood pressure more than placebo

Structure of a hypothesis test

Given our H0 and H1

Assume that H0 is true

H0: Treatment with substance X has no effect on blood pressure compared to placebo (μX = μplacebo)

1) We collect data from a scientific study.

2) Calculate the likelihood/probability/chance that data would actually turn out as it did (or even more extreme) if H0 was true;

P-value:

Probability to get at least as extreme as what we observed given H0

P-value:

Probability to get at least as extreme as what we observed given H0

P-value small:

Correct: our data is unlikely given the H0. We don’t believe in unlikely things so something is wrong. Reject H0.

Common language: It is unlikely that H0 is true; we should reject H0

P-value large:

We can´t rule out that H0 is true; we can´t reject H0

A statistical test can either reject (prove false) or fail to reject (fail to prove false) a null hypothesis, but never prove it true (i.e., failing to reject a null hypothesis does not prove it true).

μX μplacebo

μX = μplacebo

How to interpret the p-value? Cont.

From the assumption that if the treatment does not have effect we calculate: How likely is it to get the data we collected in the study?

Expected outcome of the study

for some H1 (μX < μplacebo )

Unlikely outcome if the

treatments are equal:

The p-value is small

Expected outcome of the study if

H0 is true (μX = μplacebo )

Likely outcome if the treatments

are equal:

The p-value is big

Risks of making incorrect decisions

We can make two types of incorrect decisions:

- Incorrect rejection of a true H0
“Type I error”; False Positive

2) Fail to reject a false null hypothesis H0

“Type II error”; False Negative

These two errors can´t be completely avoided.

We can try to control and minimize the risks of making

error is several ways:

1) Optimal study design

2) Deciding the burden of evidence in favor of H1

required to reject H0

Easy to misinterpret and mix things up

(p-value, size of test, significance level, etc).

P-value is not the probability that the null hypothesis

is true

P-value says nothing about the size of the effect.

A statistical significant difference does not need to be

clinically relevant!

A high p-value is not evidence that H0 is true

Distribution of average effect if Ho is true

Example:

H0: Treatment with substance X has no effect on blood pressure compared to placebo

p=0.0156

Difference in BP

Xobs=10.4mmHg

0

- If the null hypothesis is true it is unlikely to observe 10.4
- We don’t believe in unlikely things so something is wrong
- The data is ”never” wrong
- The null hypothesis must be wrong!

Distribution of average effect if Ho is true

H0: Treatment with substance X has no effect on blood pressure compared to placebo

p=0.35

Difference in BP

Xobs=3.2mmHg

0

- If the null hypothesis is true it is not that unlikely to observe 3.2
- Nothing unlikely has happened
- There is no reason to reject the null hypothesis
- This does not prove that the null hypothesis is true!

A confidence set is a random subset covering the true parameter value with probability at least .

Let

(critical function)

Confidence intervalsConfidence set:

The set of parameter values correponding to hypotheses that can not be rejected.

Example covering the true parameter value with probability at least .

- We study the effect of treatment X on change in DBP from baseline to 8 weeks
- The true treatment effect is:
d = μX – μplacebo

- The estimated treatment effect is:
- How reliable is this estimation?

Hypothesis testing covering the true parameter value with probability at least .

- Test the hypothesis that there is no difference between treatment X and placebo
- Use data to calculate the value of a test function: High value -> reject hypothesis, since it is unlikely given not true difference
- For a long time it was enough to present the point estimate and the result of the test
- Previous example: The estimated treatment effect is 7.5 mmHg, which is significantly different from zero at the 5% level.

Famous (?) quotes covering the true parameter value with probability at least .

“We do not perform an experiment to find out if two varieties of wheat or two drugs are equal. We know in advance, without spending a dollar on an experiment, that they are not equal” (Deming 1975)

“Overemphasis on tests of significance at the expense especially of interval estimation has long been condemned” (Cox 1977)

“The continued very extensive use of significance tests is alarming” (Cox 1986)

Estimations + Interval covering the true parameter value with probability at least .

Better than just hypothesis test

- The estimation gives the best guess of the true effect
- The confidence interval gives the accuracy of the estimation, i.e. limits within which the true effect is located with large certainty
- One can decide if the result is statistically significant, but also if it has biological/clinical relevance
- Now required by Regulatory Authorities, journals, etc

Neyman, Jerzy (1937). "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability". Philosophical Transactions of the Royal Society of London. Series A.

Model: covering the true parameter value with probability at least .

Drug X Placebo

σ

σ

μXμplacebo

Example- The 95% confidence interval is [-9.5, -4.9] mmHg (based on a model assuming normally distributed data)
- With great likelihood (95%), the interval [-9.5, -4.9] contains the true treatment effect μX – μplacebo
- Provided that our model is correct…

-9.5

-4.5

0

General principle covering the true parameter value with probability at least .

- There is a True fixed value that we estimate using data from a clinical trial or other experiment
- The estimate deviates from the true value (by an unknown amount)
- If we were to repeat the study a number of times, the different estimates would be centred around the true value
- A 95% confidence interval is an interval estimated so that if the study is repeated a number of times, the interval will cover the true value in 95% of these

General principle covering the true parameter value with probability at least .

- Estimate a parameter, for example
- Mean difference,
- Relative risk reduction,
- Proportion of treatment responders, etc

- Calculate an interval that contains the true parameter value with large probability

General principle covering the true parameter value with probability at least .

Depends on confidence level

Estimate

Quantile

Variability

±

*

Of the estimate

Model

General principle covering the true parameter value with probability at least .– Normal model

67% confidence level

Mean

1

±

*

SD of the mean

Normal model, σ known

General principle covering the true parameter value with probability at least .– Normal model

95 % confidence level

Mean

1.96

±

*

SD of the mean

Normal model, σ known

General principle covering the true parameter value with probability at least .– Normal model

(1-α)% confidence level

Mean

t(1-α/2, n-1)

±

*

SDof the mean

Normal model, σnot known

Width covering the true parameter value with probability at least .

- The width of the confidence interval depends on:
- Degree of Confidence (90%, 95% or 99%)
- Size of the sample
- Variability
- Statistical model
- Design of the study

Effect of confidence level covering the true parameter value with probability at least .

(normal model, n=30)

“Excuse me, professor. Why a 95% confidence interval and not a 94% or 96% interval?”, asked the student.

“Shut up,” he explained.

Effect of Sample size and SD covering the true parameter value with probability at least .

Sample size

Double SD double width, etc

Model Dependence covering the true parameter value with probability at least .

- Can we assume normally distributed data?
- Is the variability the same in all treatment groups?
- Are there important covariates, e.g. baseline level or confounders?
- Can we assume a covariance structure e.g. repeated measurements per individual?
- For time to event data – can we assume proportional hazards?
- Multiple events data – can we assume that there is no overdispersion?

Model Dependence covering the true parameter value with probability at least .

- Statistical packages provides confidence intervals for most models
- But assumptions are not always easily checked and you usually have to pre-specify the model
- Given a sufficient sample size you may still ‘get away with it’, since mean values are approximately normally distributed (or a log transform might do the trick)
- Otherwise you may want to use a non-parametric method which does not depend on a (possibly approximate) model

Non-parametric CIs - Example covering the true parameter value with probability at least .

Calculated using the available values. Example - say we have calculated the following the following treatment differences (in sorted order):

median

96.5%

88.2%

99.2%

Confidence level (interval for the median)

Analogy with tests covering the true parameter value with probability at least .

- For most statistical tests there is a corresponding confidence interval
- If a 95% confidence interval does not cover ‘the zero’, the corresponding test is to give a p-value < 0.05.
- If for example your p-value is based on the median and the confidence interval is based on the mean you can get conflicting results = trouble

Comparing two CIs covering the true parameter value with probability at least .

- Can we judge whether groups are significantly different depending on whether or not their confidence intervals overlap? - Not always.
- Parallel group:
- Non-overlapping confidence intervals - significantly different.
- Overlapping confidence intervals - not certain.
- Depends on the confidence level of the individual CIs: Comparing two 84% confidence intervals corresponds to a test at the 5% level (approximately)

Sample size covering the true parameter value with probability at least .

- Based on the precision of the interval instead of the power of a test
- Examples:
- Interval of treatment difference not wider than x, with some probability
- Lower limit/mean > 0.6 and upper limit/mean < 1.4 with at least 80% probability* - suitable when estimating geometric means
* Based on ‘FDA recommendation’ for pediatric PK studies

Example covering the true parameter value with probability at least .

Objective: To compare sitting diastolic blood pressure (DBP) lowering effect of hypersartan 16 mg with that of hypersartan 8 mg

Variable: The change from baseline to end of study in sitting DBP

(sitting SBP) will be described with an ANCOVA model,

with treatment as a factor and baseline blood pressure

as a covariate

treatment effect

i = 1,2,3

{16 mg, 8 mg, 4 mg}

yij = μ + τi + β(xij - x··) + εij

Model:

Null hypoteses (subsets of ):

H01: τ1 = τ2 (DBP)

H02: τ1 = τ2 (SBP)

H03: τ2 = τ3 (DBP)

H04: τ2 = τ3 (SBP)

Parameter space:

Example contined covering the true parameter value with probability at least .

This is a t-test where the test statistic follows a t-distribution

Rejection region:

0

-c

c

P-value: The null hypothesis can pre rejected at

0

-4.6

-2.8

P-value says nothing about the size of the effect! covering the true parameter value with probability at least .

Example: Simulated data. The difference between treatment and placebo is 0.3 mmHg

A statistical significant difference does NOT need to be clinically relevant!

Statistical and clinical significance covering the true parameter value with probability at least .

Is there any difference between the evaluated treatments?

Statistical significance:

Clinical significance:

Does this difference have any meaning for the patients?

Is there any economical benefit for the society in using the new treatment?

Health ecominical relevance:

Statistical and clinical significance covering the true parameter value with probability at least .

A study comparing gastroprazole 40 mg and mygloprazole 30 mg with respect to healing of erosived eosophagitis after 8 weeks treatment.

Cochran Mantel Haenszel p-value = 0.0007

Statistically significant!

Clinically significant?

Health economically relevant??

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