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 Concept of standard deviation vs standard error and limits of confidence /confidence interval 2

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 3

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

Inferential statistics Uses patterns in the sample data to draw inferences about the population represented, accounting for randomness. Two basic approaches: – Hypothesis testing – Estimation Common goal: conclude on the effect of an independent variable (exposure) on a dependent variable (outcome). 5

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. 6

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? 7

The two hypothesis! There is a difference between the two groups (=there is an effect) Alternative Hypothesis (H 1 ) (e.g.: RR=3.6) When you perform a test of statistical significance you usually reject or do not reject the Null Hypothesis (H 0 ) There is NO difference between the two groups (=no effect) Null Hypothesis (H 0 ) (e.g.: RR=1) 8

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.” 9

Hypothesis testing and null hypothesis Tests of statistical significance Data not consistent with H 0 : – H 0 can be rejected in favour of some alternative hypothesis H 1 (the objective of our study). Data are consistent with the H 0 : – H 0 cannot be rejected You cannot say that the H 0 is true. You can only decide to reject it or not reject it. 10

How to decide when to reject the null hypothesis? H 0 rejected using reported p value 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 11

p values – practicalities Small p values = low degree of compatibility between H 0 and the observed data:  you reject H 0, the test is significant Large p values = high degree of compatibility between H 0 and the observed data:  you don’t reject H 0, the test is not significant We can never reduce to zero the probability that our result was not observed by chance alone 12

Levels of significance – practicalities We need a cut-off ! p value > 0.05 = H 0 non rejected (non significant) p value ≤ 0.05 = H 0 rejected (significant) BUT: Give always the exact p-value rather than „significant“ vs. „non-significant“. 13

”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 relationship was statistically significant (p=0.0361)” Examples from the literature 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) 14

p = 0.05 and its errors Level of significance, usually p = 0.05 p value used for decision making But still 2 possible errors: H 0 should not be rejected, but it was rejected :  Type I or alpha error H 0 should be rejected, but it was not rejected :  Type II or beta error 15

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

H 0 is “true” but rejected: Type I or  error H 0 is “false” but not rejected: Type II or  error Types of errors Decision based on the p value Truth No diff Diff 17

More on errors Probability of Type I error: – Value of α is determined in advance of the test – The significance level is the level of α error that we would accept (usually 0.05) Probability of Type II error: – Value of β depends on the size of effect (e.g. RR, OR) and sample size – 1-β: Statistical power of a study to detect an effect on a specified size (e.g. 0.80) – Fix β in advance: choose an appropriate sample size 18

 H 0 is true H 1 is true Test statistics T  Even more on errors 19

Principles of significance testing Formulate the H 0 Test your sample data against H 0 The p value tells you whether your data are consistent with H 0 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 H 0, or fail to reject it! 20

11.21 Interpreting the p-value… Overwhelming Evidence (Highly Significant) Strong Evidence (Significant) Weak Evidence (Not Significant) No Evidence (Not Significant) p=.0069

11.22 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!

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)]=√SD 1 / n 1 + SD 2 /n 2 4. SE of difference between two proportions= √p 1 q 1 /n 1 + p 2 q 2 /n 2

11.24 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: H 0 : — the ‘null’ hypothesis H 1 : — the ‘alternative’ or ‘research’ hypothesis The null hypothesis (H 0 ) will always state that the parameter equals the value specified in the alternative hypothesis (H 1 ) pronounced H “nought”

11.25 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

We rephrase the hypothesis as Null = the mean is = 350 H 0 : = 350 Thus, our research hypothesis becomes: H 1 : ≠ 350 While sd = 75, sample size = 25 and Sample mean calculated to be

11.27 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.

11.28 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 hypothesis in 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…

11.29 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: H 0 : = 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 /(25) = 15

11.30 Is the Sample Mean in the Guts of the Sampling Distribution??

11.31 Three ways to determine this: First way 1.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 = *15 = = LCV = 350 – 1.96*15 = 350 – 29.4 = 320.6

Unstandardized Test Statistic Approach

11.33 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 = ( - )/ = ( – 350)/15 = Is this Z-Score in the guts of the sampling distribution???

Standardized Test Statistic Approach

11.35 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( > ) = P(Z > 1.344) = Since this is a two tailed test, you must double this area for the p-value. p-value = 2*(0.0901) = 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].

p-value approach

11.37 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.

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. 38

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 39

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 measure 95% of the time.” 40

Confidence interval (CI) Indicates the amount of random error in the estimate Can be calculated for any „test statistic“, e.g.: means, proportions, ORs, RRs e.g. CI for means 95% CI = x – 1.96 SE up to x SE 1 - α α/2 Lower limit upper limit of 95% CI  = 5% s 41

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

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%) Width of confidence interval depends on … 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 43

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

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”. 45

Comments on p values and CIs A huge effect in a small sample or a small effect in a large sample can result in identical p values. 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. 46

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