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Lecture 7,8

Lecture 7,8

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Lecture 7,8

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  1. Lecture 7,8 Summarizing Data, hypothesis testing and expression of results

  2. Purpose of Statistics • To describe & summarize information, thereby reducing it to smaller, more meaningful sets of data • To make predictions or to generalize about occurrences based on observations • To identify associations, relationships or differences between sets of observations

  3. Main Types of Statistics • Descriptive Statistics • Inferential Statistics

  4. Descriptive Statistics • Descriptive Statistics involves organizing, summarizing & displaying data to make them more understandable. • Most common statistics used are frequencies, percents, measures of central tendency, summary tables, charts & figures.

  5. Inferential Statistics • Inferential Statistics: a set of statistical techniques that provides predictions about population characteristics based on information from a sample drawn from that population. • Inferential statistics report the degree of confidence of the sample statistic that predicts the value of the population parameter.

  6. Describing data

  7. Describing data

  8. Measures of Central Tendency • When assessing the central tendency of your measurements, you are attempting to identify the “average” measurement • Mean:best known & most widely used average, describing the center of a frequency distribution • Median: the middle value/point of a set of ordered numbers below which 50% of the distribution falls • Mode: the most frequent value or category in a distribution

  9. Thinking Challenge $400,000 $70,000 $50,000 ... employees cite low pay -- most workers earn only $20,000. ... President claims average pay is $70,000! $30,000 $20,000

  10. Comparison of Central Tendency Measures • Use Mean when distribution is reasonably symmetrical, with few extreme scores and has one mode. • Use Median with nonsymmetrical distributions because it is not sensitive to skewness. • Use Mode when dealing with frequency distribution for nominal data

  11. Variability • A quantitative measure of the degree to which scores in a distribution are spread out or are clustered together; • Types of variability include: • Standard Deviation: a measure of the dispersion of scores around the mean • Range: Highest value minus the lowest value • Interquartile Range: Range of values extending from 25th percentile to 75th percentile

  12. Key Principles of Statistical Inference

  13. Explaining precision or certainty of effect • Studies can only ever hazard a guess at the effect of exposures or interventions in the target population.

  14. Explaining precision or certainty of effect • Imagine you tested a tablet on 100 adults to see whether it reduced arthritic pain. You find that it reduces pain in 63 of 100 people. That is, it has an absolute risk reduction of 63%. Say you now randomly sampled 100 different adults, and repeated exactly the same experiment. Do you think 63 of 100 would have pain relief? Probably not. By chance, this time you might find that only 58 of the 100 people had pain relief. This second study would have, therefore, reported an absolute risk reduction of 58%. If you repeated the experiment enough times, you might, by chance, get a few really weird results, like 30% or 88%, which were a long way away from the rest.

  15. Explaining precision or certainty of effect • Luckily, there is a law of statistics which begins to get around this difficulty. • It says that if you were to conduct more and more studies on random samples from the target population and keeping averaging out their results, you will get closer and closer to the ‘Truth.’

  16. Explaining precision or certainty of effect • This is all well and good if you can average out results from lots of studies (this is what meta-analyses do – providing they are carried out properly). • But what if there are only one or two studies? Fortunately, the second part of the law says that the results from different studies are distributed in a predictable way (in a normal or Gaussian distribution) around the ‘True’ value. So, even from a single study, we can predict what the results of other, similar studies might be, were we to conduct them.

  17. Explaining precision or certainty of effect • Statisticians can, therefore, use this aspect of the law to calculate confidence intervals around a single study result . Thus, the confidence interval may be accurately interpreted in the following way: • The range of values in which you would expect a specified percentage (usually 95%) of study results from studies like this one to fall, had you conducted them.

  18. CENTRAL LIMIT THEOREM • A. For a randomly selected sample of size n (n should be at least 25, but the larger n is, the better the approximation) with a mean u and a standard deviation o-. • _ • 1.                    The distribution of sample means x is approximately normal regardless of whether the population distribution is normal • From statistical theory come these two additional theories • 2.                    The mean of distribution of sample means is equal to the mean of the population distribution – that is

  19. 3. The standard deviation of the distribution of sample means calculated by the usual methods is very close to the standard error of the population mean calculated by the standard deviation of the population divided by the square root of the sample size – • ẍ = SEẍ = /sr n

  20. STANDARD ERROR OF THE MEAN The measure of variation of the distribution of sample means, , referred to as the standard error of the mean, is denoted as - that is

  21. POINT ESTIMATES AND CONFIDENCE INTERVALS

  22. Important note • We are confident that 95% of samples mean are within 1.96 sd of the population mean and then 95% of all samples will provide A CI that capture the actual population mean

  23. And • We are confident that 95% of samples mean differences are within 1.96 sd of the population mean difference and then 95% of all samples mean differences will provide A CI that capture the actual population mean difference

  24. Population estimates AND Confidence interval • If we have a sample of 25 people and their average blood glucose is 161.84 ( equals 58.15) what is the 95% CI of mean= =sample mean±1.96SE= 138.6-185.24 (upper and lower confidence limit)

  25. Interpretation of the confidence interval • The upper and lower limits of the confidence interval can be used to assess whether the difference between the two mean values is clinically important. • For example, if the lower limit is close to zero, this indicates that the true difference may be very small and clinically meaningless, even if the test is statistically significant.

  26. Significance Testing

  27. Null Hypothesis - Ho • Ho proposes no difference or relationship exists between the variables of interest • Foundation of the statistical test • When you statistically test an hypothesis, you assume that Ho correctly describes the state of affairs between the variables of interest

  28. Example (Hypothesis testing using p value) • Drug A= 95±12.4 (n=10) • Drug B=105±11.8 (n=10) • Null hypothesis is that mean difference=0 • alpha=0.05 • Mean difference =10 • SE for the difference=5.4

  29. SPSS Output for Two Sample Independent t-test Example

  30. What does this mean • 1. the difference is due to chance • 2. the samples came from a population with a true mean difference that is equal to 0 • 3. mean difference of 10 has a probability of 8%

  31. P-values • The p value measures how likely a particular difference between groups is to be due to chance. • A p-value of 0.1 means that, extrapolating from the results of our study, there is a 10% chance that in ‘Truth,’ there is no effect. • A p-value of 0.05 means that there is a 5% chance the in ‘Truth’ there is no effect. • Usually, we studies define p-values of <0.05 as being statistically significant. There is nothing magic about this cut-off. What the authors are saying when they choose this cut-off is that they are happy to report that there is an effect, providing the chance that they are wrong (ie in ‘Truth’ there is no effect) is less than 5%.

  32. Hypothesis testing Using CI • Mean difference =10 • SE=5.4 • CI=2*5.4 • 10±10.8= -0.2 to 20.8 • Accept the null hypothesis • P>0.05

  33. Confidence Interval • The range of the actual difference between the two drugs • Mean difference in HBA1c reduction 1 (0.1-1.9) • Absolute risk reduction 30% (3-70%) • NNT 3 (1-10000) • RRR 40% (20-120)

  34. Interpretation of the confidence interval • The upper and lower limits of the confidence interval can be used to assess whether the difference between the two mean values is clinically important. • For example, if the lower limit is close to zero, this indicates that the true difference may be very small and clinically meaningless, even if the test is statistically significant.

  35. SPSS Output for Two Sample Independent t-test Example

  36. Confidence intervals show a statistically significant result if they “do not cross the line of no effect.” What does this mean? Basically, you can tell whether a result is statistically significant by seeing whether the confidence intervals include zero (for comparative absolute measures of risk, such as absolute risk reduction) or one (for relative measures of risk, including relative risks and odds ratios).

  37. Statistical Significance VSMeaningful Significance • Common mistake is to confuse statistical significance with substantive meaningfulness • Statistically significant result simply means that if Ho were true, the observed results would be very unusual • With N > 100, even tiny relationships/differences are statistically significant

  38. Statistical Significance VSMeaningful Significance • Statistically significant results say nothing about clinical importance or meaningful significance of results • Researcher must always determine if statistically significant results are substantively meaningful. • Refrain from statistical “sanctification” of data

  39. Confidence Interval (0.3 – 4)Lower limit must be > 2 to be Clinical significant

  40. HOW THE RESULTS ARE EXPRESSED Understanding Results

  41. Mean Difference • Drug A reduced HBA1c by 1% • Drug B reduced HBA1c by 2% • Mean difference =

  42. The benefits or harms of a treatment can be shown in various ways: • Drug X produced an absolute reduction in deaths by 7 per cent ("absolute risk reduction") • Drag X reduced the death rate by 28 per cent ("relative risk reduction") • Drug X increased the patients' survival rate from 75 to 82 percent • 14 people would need to be treated with drug X to prevent one death ("number needed to treat")

  43. death 8 weekTreatment No death New Drug 999999999 1 1000000000 a b c d Old drug 999999998 2 1000000000 200000

  44. Esophageal bleeding 8 weekTreatment No bleeding Proton Pump Inhibitor 45 55 100 a b c d H2 Antagonist 75 25 100 200 80 120 GERD Treatment: 2 x 2 Table

  45. Relative Risk (RR) • The relative risk of an outcome is the chances of that outcome occurring in the treatment group compared with the chances of it occurring in the control group. • If the chances are the same in both groups, the relative risk is 1. • The relative risk reduction (RRR) is the amount by which the risk (of death) is reduced by drug X as a comparative percentage of the control, calculated as: