Lecture 6 Statistical Tests. Confidence intervals Student test ANOVA test Fisher test for variances Nonparametric tests

Transcription

1 Lecture 6 Statistical Tests Confidence intervals Student test ANOVA test Fisher test for variances Nonparametric tests

2 Gauss Curve Average=100, Standard deviation=15

3 Sample from a normal population In the following 9 images you will see: In the first one the distribution of values for a Gauss population In the second - the distribution of averages computed for groups of two individuals In the next images, the distribution of averages computed for groups of 3, 4, 9, 16, 25, 36 and 100 individuals

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5 Conclusion: Averages calculated on a sample of size n drawn from a Gaussian distributed population have: A Gaussian distribution An average equal to the average of the population of origin Standard deviation equal to the deviation of the population divided by square root of n= STANDARD ERROR! Err = σ / n

6 Confidence interval- definition We call confidence interval for the mean an interval of real numbers where we are almost certain that the real mean is located. The level of confidence can be chosen; usually it is 95% or 99% P m t c s n m t c s n 0,95

7 Example: CI 95% Values: Latencies measured over the optic nerve Average: 112,2 Standard deviation: 12,5 Number of cases: 156 Standard error: Coefficient used for 95% confidence: The limits of the interval are: => CI 95% =[110.24; 114,16]

8 What is a statistical test? It is a decision method that helps us to validate or invalidate a statistical hypothesis with a certain degree of confidence.

9 Statistical tests Statistical tests verify the truthfulness of statistical hypotheses => a process called statistical inference hypothesis H0 (or null hypothesis) data have no connections between them/ compared values do not differ hypothesis H1 (or alternate hypothesis) data have connections between them/ compared values do differ

10 Statistical tests The result of the test is denoted as p. p is a number between 0 and 1 (or 0% and 100%), and it represents the probability to make an error if we reject H0, the null hypothesis. Interpreting p (the same for all statistical tests): p > 0.05, the statistical relationship is not significant (NS). p < 0.05, the statistical relationship is significant (S, 95% confidence). p < 0.01, the statistical relationship is significant (S, 99% confidence). p < 0.001, the statistical relationship is highly significant (HS, 99.9% confidence).

11 Student t test Test for comparing two means when the standard deviations are equal It can be used if: 1. measurements from the two samples are independent 2. the samples come from populations that are normally distributed (which must be verified before applying the test) 3. the populations from which the samples are drawn have equal dispersions (or standard deviations)

12 Student t test We call H0 the null hypothesis - the assumption that the averages of the populations (from which the samples are drawn) are equal -----H 0 : m 1 = m 2 We call H1 the alternate hypothesis - the assumption that the averages of the populations are different H 1 : m 1 m 2 If the test doesn t reject H 0, we say that data do not support the hypothesis that the populations means are different If the test rejects H 0, we say that data support the hypothesis that the populations means are different

13 Example We analyzed 25 patients with localised infection and 25 with sepsis, among other measurements we recorded ESR value. Pacient No. Localised infection Sepsis Media Deviaţia standard Dispersia C.V. (%) 44.67% 34.81% Mean ± standard deviation t Test: Two Sample Assuming Equal Variances Localised infection ESR p<0,05 we reject H 0 with a confidence of 95% Sepsis p test Student = S Localised infection Sepsis EXPLANATIONS Mean Samples averages Variance Samples variance Observations Number of patients Pooled Variance Hypothesized Mean Difference 0 df 48 Degrees of freedom: t Stat Computed t value P(T<=t) one tail t Critical one tail P(T<=t) two tail p test result t Critical two tail Theoretical t value)

14 Confidence interval analysis Parameter Localised infection Sepsis No. of patients Mean value Standard deviation Standard error t 95% for df= Error level for 95% conf Lower limit Upper limit We notice that the upper limit of the 95% confidence interval for the lower average is below the lower limit of the 95% for the higher average. In conclusion, we can say that the two averages are different with a confidence level of 95% - which we did using Student s t test.

15 Student s t test - variants t test for samples with equal variances t test for samples with unequal variances t test for samples with paired measurements These variants differ in how the coefficient t is calculated, and thus the p-value.

16 ANOVA test Compares the averages of several samples simultaneously. H0: m1 = m2 = m3 = m4 (for 4 samples) H1: at least two averages are significantly different The result is a number p that is interpreted as follows: If p> 0.05 we do not reject H0, we say that the difference is not significant with 95% confidence If p <0.05 H0 is rejected with a confidence level of 95% - At least two averrages are significantly different If p <0.001 H0 is rejected with a confidence level of 99.9%. The difference is highly significant

17 Exemplu In three cities from Dolj county data were taken on dietary habits and their relationship to obesity and diabetes. Among other data we collected the weight of individuals as well as data on smoking Individuals, regardless of gender or age group, were divided into four categories: non-smokers, former smokers, light smokers (less than 10 cigarettes per day) and smokers (more than 10 cigarettes per day) An interesting question is whether there was a link between smoking habit and body weight in these individuals. H0: Whether or not smoking, body weight is the same H1: At least two of the four categories had different body weights

18 ANOVA test - results

19 Variance comparison tests Fischer s test is used to verify the equality of variances for two normally distributed independent variables. Null hypothesis is H 0 : σ 12 =σ 2 2 Bartlett's test is used to verify the equality of variances for several normally distributed independent variables Null hypothesis is H 0 : σ 12 =σ 22 =...=σ k 2 THEY ARE PARAMETRIC TESTS

20 Nonparametric tests These are tests that do not make assumptions about data distribution. They can be applied in all circumstances, if the measurements are independent Usually they are used when we are not allowed to apply a parametric test These tests have less accurate results than parametric tests, so we apply them only when it is not possible to apply a parametric test

21 Nonparametric tests Comparing means Random data Paired data 2 samples Mann-Whitney Wilcoxon >2 samples Kruskal-Wallis Friedman Comparing variances Levene