1 / 47

Significance testing and confidence intervals

Significance testing and confidence intervals. Col Naila Azam. RECAP OF BIOSTATISTICS. Types of data, variables, census ,research and distribution of data( normal, skewed) Description of data; measures of central tendency ,dispersion and implication in describing various types of data

nessa
Télécharger la présentation

Significance testing and confidence intervals

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Significance testingand confidence intervals Col NailaAzam

  2. RECAP OF BIOSTATISTICS • Types of data, variables, census ,research and distribution of data( normal, skewed) • Description of data; measures of central tendency ,dispersion and implication in describing various types of data • Concept of standard deviation vs standard error and limits of confidence /confidence interval

  3. Learning objectives • To understand the concept of hypothesis formulation • To delineate the implications of various types of hypothesis(one tailed/two tailed) on research results in epidemiology • To grasp the concept of p – value in result interpretation • To clarify the concept of critical value in data distribution and the possible errors

  4. The idea of statistical inference Generalisation to the population Conclusions based on the sample Population Hypotheses Sample

  5. Inferential statistics • Usespatterns in the sample data to draw inferences about the population represented, accounting for randomness. • Twobasicapproaches: • Hypothesis testing • Estimation • Commongoal: concludeontheeffect of an independentvariable (exposure) on a dependentvariable (outcome).

  6. The aim of a statistical test To reach a scientific decision (“yes” or “no”) on a difference (or effect), on a probabilistic basis, on observed data.

  7. Why significance testing? Gastroenteritis outbreak in karachi: “The risk of illness was higher among diners who ate home preserved pickles (RR=3.6).” Is the association due to chance?

  8. The two hypothesis! When you perform a test of statistical significance you usually reject or do not reject the Null Hypothesis (H0)

  9. gastroenteritis outbreak in karachi • Null hypothesis (H0): “There is no association between consumption of green pickles and gastroenteritis.” • Alternative hypothesis(H1): “There is an association between consumption of green pickles and gastroenteritis.”

  10. Hypothesis testing and null hypothesis • Tests of statistical significance • Data not consistent with H0 : • H0 can be rejected in favour of some alternative hypothesis H1 (the objective of our study). • Data are consistent with the H0: • H0 cannot be rejected You cannot say that the H0 is true. You can only decide to reject it or not reject it.

  11. How to decide when to reject the null hypothesis? H0 rejected using reportedpvalue p-value = probability that our result (e.g. a difference between proportions or a RR) or more extreme values could be observed under the null hypothesis

  12. p values – practicalities Small p values = low degree of compatibility between H0 and the observed data: you reject H0, the test is significant Large p values= high degree of compatibility between H0 and the observed data: you don’t reject H0, the test is not significant We can never reduce to zero the probability that our result was not observed by chance alone

  13. Levels of significance – practicalities We need a cut-off ! 0.01 0.05 0.10 p value > 0.05 = H0 non rejected (non significant) p value ≤ 0.05 = H0 rejected (significant) BUT: Give always the exact p-value rather than „significant“ vs. „non-significant“.

  14. Examples from the literature • ”The limit for statistical significance was set at p=0.05.” • ”There was a strong relationship (p<0.001).” • ”…, but it did not reach statistical significance (ns).” • „ The relationshipwasstatisticallysignificant (p=0.0361)” p=0.05 Agreed convention Not an absolute truth ”Surely, God loves the 0.06 nearly as much as the 0.05” (Rosnow and Rosenthal, 1991)

  15. p = 0.05 and its errors • Level of significance, usually p = 0.05 • p value used for decision making But still 2 possible errors: • H0should not be rejected, but it was rejected : • Type I or alpha error • H0should be rejected, but it was not rejected : Type II or beta error

  16. Keep in mind that committing a Type I error OR a Type II error can be VERY bad depending on the problem.

  17. Types of errors Truth No diff Diff Decision based on the p value No diff Diff • H0 is “true” but rejected: Type I or  error • H0 is “false” but not rejected: Type II or  error

  18. More on errors • Probability of Type I error: • Value of α is determinedinadvance of the test • The significancelevel is thelevel of αerrorthatwewouldaccept (usually 0.05) • Probability of Type II error: • Value of βdependsonthesize of effect (e.g. RR, OR) and samplesize • 1-β: Statisticalpower of a studytodetect an effecton a specifiedsize (e.g. 0.80) • Fix βinadvance: choose an appropriatesamplesize

  19. Even more onerrors H1 is true H0 is true b a Test statistics T

  20. Principles of significance testing • Formulate the H0 • Test your sample data against H0 • The p value tells you whether your data are • consistent with H0 • i.e, whether your sample data are consistent with a chance finding (large p value), or whether there is reason to believe that there is a true difference (association)between the groups you tested • You can only reject H0, or fail to reject it!

  21. Interpreting the p-value… Overwhelming Evidence (Highly Significant) Strong Evidence (Significant) Weak Evidence (Not Significant) No Evidence (Not Significant) 0 .01 .05 .10 p=.0069

  22. Conclusions of a Test of Hypothesis… • If we reject the null hypothesis, we conclude that there is enough evidence to infer that the alternative hypothesis is true. • If we fail to reject the null hypothesis, we conclude that there is not enough statistical evidence to infer that the alternative hypothesis is true. This does not mean that we have proven that the null hypothesis is true!

  23. FORMULAE FOR ESTIMATION OF STANDARD ERROR(SE) OF SAMPLE • 1. SE of sample mean= SD/ √n • 2. SE of sample proportion(p) = √pq/n • 3. SE of difference between two means[SE(d)]=√SD1/ n1+ SD2/n2 • 4. SE of difference between two proportions= √p1q1/n1+ p2q2/n2

  24. Concepts of Hypothesis Testing (1)… • The two hypotheses are called the null hypothesis and the other the alternative or research hypothesis. The usual notation is: • H0: — the ‘null’ hypothesis • H1: — the ‘alternative’ or ‘research’ hypothesis • The null hypothesis (H0) will always state that the parameter equals the value specified in the alternative hypothesis (H1) pronounced H “nought”

  25. Concepts of Hypothesis Testing… • Consider mean demand for computers during assembly lead time. Rather than estimate the mean demand, our operations manager wants to know whether the mean is different from 350 units. In other words, someone is claiming that the mean time is 350 units and we want to check this claim out to see if it appears reasonable. • Recall that the standard deviation [σ]was assumed to be 75, the sample size [n] was 25, and the sample mean [ ] was calculated to be 370.16

  26. We rephrase the hypothesis as • Null = the mean is = 350 • H0: = 350 • Thus, our research hypothesis becomes: • H1: ≠ 350 While sd = 75, sample size = 25 and • Sample mean calculated to be 370.6

  27. Concepts of Hypothesis Testing… • For example, if we’re trying to decide whether the mean is not equal to 350, a large value of (say, 600) would provide enough evidence. • If is close to 350 (say, 355) we could not say that this provides a great deal of evidence to infer that the population mean is different than 350.

  28. Concepts of Hypothesis Testing (4)… • The two possible decisions that can be made: • Conclude that there is enough evidence to support the alternative hypothesis (also stated as: reject the null hypothesisin favor of the alternative) • Conclude that there is not enough evidence to support the alternative hypothesis (also stated as: failing to reject the null hypothesis in favor of the alternative) NOTE: we do not say that we accept the null hypothesis if a statistician is around…

  29. Concepts of Hypothesis Testing (2)… • The testing procedure begins with the assumption that the null hypothesis is true. • Thus, until we have further statistical evidence, we will assume: • H0: = 350 (assumed to be TRUE) • The next step will be to determine the sampling distribution of the sample mean assuming the true mean is 350. • is normal with 350 • 75/(25) = 15

  30. Is the Sample Mean in the Guts of the Sampling Distribution??

  31. Three ways to determine this: First way • Unstandardized test statistic: Is in the guts of the sampling distribution? Depends on what you define as the “guts” of the sampling distribution. • If we define the guts as the center 95% of the distribution [this means  = 0.05], then the critical values that define the guts will be 1.96 standard deviations of X-Bar on either side of the mean of the sampling distribution [350], UCV = 350 + 1.96*15 = 350 + 29.4 = 379.4 • LCV = 350 – 1.96*15 = 350 – 29.4 = 320.6

  32. 1. Unstandardized Test Statistic Approach

  33. Three ways to determine this: Second way • 2. Standardized test statistic: Since we defined the “guts” of the sampling distribution to be the center 95% [ = 0.05], • If the Z-Score for the sample mean is greater than 1.96, we know that will be in the reject region on the right side or • If the Z-Score for the sample mean is less than -1.97, we know that will be in the reject region on the left side. Z = ( - )/ = (370.16 – 350)/15 = 1.344 • Is this Z-Score in the guts of the sampling distribution???

  34. 2. Standardized Test Statistic Approach

  35. Three ways to determine this: Third way • 3. The p-value approach (which is generally used with a computer and statistical software): Increase the “Rejection Region” until it “captures” the sample mean. • For this example, since is to the right of the mean, calculate • P( > 370.16) = P(Z > 1.344) = 0.0901 • Since this is a two tailed test, you must double this area for the p-value. • p-value = 2*(0.0901) = 0.1802 • Since we defined the guts as the center 95% [ = 0.05], the reject region is the other 5%. Since our sample mean, , is in the 18.02% region, it cannot be in our 5% rejection region [ = 0.05].

  36. 3. p-value approach

  37. Statistical Conclusions: • Unstandardized Test Statistic: • Since LCV (320.6) < (370.16) < UCV (379.4), we fail to reject the null hypothesis at a 5% level of significance. Standardized Test Statistic: • Since -Z/2(-1.96) < Z(1.344) < Z/2 (1.96), we fail to reject the null hypothesis at a 5% level of significance. P-value: • Since p-value (0.1802) > 0.05 [], we fail to reject the hull hypothesis at a 5% level of significance.

  38. Criticism on significance testing “Epidemiological application need more than a decision as to whether chance alone could have produced association.” (Rothman et al. 2008) →Estimation of an effect measure(e.g. RR, OR) rather than significance testing.

  39. The epidemiologist needs measurements rather than probabilities 2 is a test of association OR, RR are measures of association on a continuous scale infinite number of possible values The best estimate = point estimate Range of values allowing for random variability: Confidence interval precision of the point estimate

  40. Confidence interval (CI) Range of values, on the basis of the sample data, in which the population value (or true value) may lie. • Frequently used formulation: „If the data collection and analysis could be replicated many times, the CI should include the true value of the measure95% of the time.”

  41. Confidenceinterval (CI) e.g. CI for means 95% CI = x – 1.96 SEuptox + 1.96 SE a = 5% 1 - α α/2 α/2 s Lower limit upper limit of 95% CI of 95% CI Indicatestheamount of random errorintheestimate Can becalculatedforany „teststatistic“, e.g.: means, proportions,ORs, RRs

  42. CI terminology Point estimate Confidence interval RR = 1.45 (0.99 – 2.1) Lower confidence limit Upper confidence limit

  43. Width of confidence interval depends on … • The amount of variability in the data • The size of the sample • The arbitrary level of confidence you desire for your study (usually 90%, 95%, 99%) A common way to use CI regarding OR/RR is : If 1.0 is included in CI  non significant If 1.0 is not included in CI  significant

  44. What we have to evaluate the study • 2A test of association. It depends on sample size. • p valueProbability that equal (or more extreme) results can be observed by chance alone • OR, RRDirection & strength of association if > 1 risk factor if < 1 protective factor (independently from sample size) • CI Magnitude and precision of effect

  45. Comments on p-values and CIs • Presence of significance does not prove clinical or biological relevance of an effect. • A lack of significance is not necessarily a lack of an effect: “Absence of evidence is not evidence of absence”.

  46. Comments on pvalues and CIs • A huge effect in a small sample or a small effect in a large sample can result in identical pvalues. • A statistical test will always give a significant result if the sample is big enough. • p values and CIs do not provide any information on the possibility that the observed association is due to bias or confounding.

  47. THANK YOU FOR APPRECIATING LOGIC OF BIOSTATISTICS

More Related