STATISTICAL TESTS USED IN VARIOUS STUDIES

BIOSTATISTICS

1. Dr Ashish PG first Yr. Dept. of Biochemistry Kcgmc karnal Statistical tests

2. TEST OF SIGNIFICANCE A test of significance is a statistical method used to determine whether the observed data in an experiment or study are strong enough to reject a null hypothesis. The null hypothesis typically represents a default assumption, such as no effect or no difference between groups. The test assesses the probability that the observed results would occur if the null hypothesis were true. This probability is represented by the p-value.

3. Confidence Interval (CI) A confidence interval gives a range of values within which we expect the true population parameter (like the mean) to fall, based on our sample data. confidence interval 95% : This tells us we are 95% confident that the interval we calculate from our sample data includes the true population parameter.( like the mean)

4. p- Value It is a standard against which we compare our result It is a result of computation Computed p-value compared with p-value criterion to test statistical significance Smaller p-value is always better p≤0.05 means reject null hypothesis and reached statistical significance p>0.05 means do not reject null hypothesis and has not reached statistical significance

5. Types of error Type I error/ alpha error- rejecting the null hypothesis when it is true (error of commission) Actually no difference, but test saying there is difference Worse than beta Asserts that drug works, when it does not Type II/ beta error- failing to reject null hypothesis when it is false (error of omission) Actually there is difference, but test saying there is no difference Asserts that drug does not work, when it really does

6. Statistical tests PARAMETRIC TESTS NON PARAMETRIC TESTS STUDENT T TEST ( amt) 1) PAIRED T TEST 2)UNPAIRED T TEST CHI-SQUARE TEST (Qu) Z TEST (amt) FISHER’S EXACT TEST ( qu ) ANOVA ( both ) KRUSKAL-WALLIS TEST ( amt ) TUKEY'S HONEST SIGNIFICANT DIFFERENCE (HSD) TEST BONFERRONI POST HOC TEST PEARSON CORRELATION (amt) SPEARMAN CORRELATION(amt) Wilcoxon signed-rank test ( amt ).

7. Paired t-Test and Unpaired t-Test Photo by Pexels

8. Paired T-test Purpose: The paired t-test is used when you have two measurements taken on the same group of subjects under different conditions or at different times. It assesses whether the mean difference between the paired observations is significantly different from zero. When to Use: When you have two related samples (e.g., before and after measurements on the same individuals). When you want to evaluate the effect of a treatment or intervention within the same group.

9. Example: Suppose a researcher wants to evaluate the effect of a new diet on weight loss. They measure the weight of 10 participants before starting the diet and then again after 6 weeks on the diet. The weights are paired because each participant is measured twice, once before and once after the diet. Before Diet (Weight in kg): 80, 85, 90, 95, 100, 105, 110, 115, 120, 125 After Diet (Weight in kg): 78, 82, 88, 93, 98, 102, 108, 112, 118, 123 In a paired t-test, you would calculate the differences between the paired weights (Before - After) for each participant and then test if the mean of these differences is significantly different from zero.

10. Paired t-test: Compares means from the same group at different times or conditions. Use it for repeated measures or matched samples. Unpaired t-test: Compares means between two independent groups. Use it when the groups are unrelated. Both tests assume that the data are normally distributed. Difference between Paired t-test and unpaired t-test

11. Paired t-test Calculation Example Data Before Diet (Weight in kg): 80, 85, 90, 95, 100, 105, 110, 115, 120, 125 After Diet (Weight in kg): 78, 82, 88, 93, 98, 102, 108, 112, 118, 123

12. Steps: Calculate the differences between the paired observations. Differences (Before - After): 80 - 78 = 2 85 - 82 = 3 90 - 88 = 2 95 - 93 = 2 100 - 98 = 2 105 - 102 = 3 110 - 108 = 2 115 - 112 = 3 120 - 118 = 2 125 - 123 = 2 Differences: 2, 3, 2, 2, 2, 3, 2, 3, 2, 2

13. * Degree of freedom in statistics reflect the number of independent values in a data set that can vary without breaking any constraints imposed by the statistical model or estimation method.

14. Z test A Z-test is a statistical test used to determine whether there is a significant difference between the means of two groups, or to compare a sample mean to a known population mean when the population variance is known and the sample size is large (typically n>30). The Z-test relies on the assumption that the data follows a normal distribution. Testing Enzyme Activity Let's say we are studying an enzyme that catalyzes a specific biochemical reaction. We know from previous studies that the average activity level of this enzyme in a healthy population is 50 units per milliliter (U/mL) with a known standard deviation of 5 U/ mL.

15. Hypothesis: We suspect that a new drug has an effect on the enzyme's activity. After administering the drug to a sample of 40 patients, we measure the enzyme activity and find an average activity of 52 U/ mL. Objective: To determine whether the observed increase in enzyme activity is statistically significant or just due to random chance, you perform a Z-test. Steps in the Z-Test: 1) Formulate Hypotheses : Null Hypothesis (H0): The mean enzyme activity after drug administration is 50 U/mL (no effect). μ=50 Alternative Hypothesis (H1​): The mean enzyme activity after drug administration is different from 50 U/mL (effect present). μ≠50

16. Determine the Critical Value: For a significance level (alpha) of 0.05 in a two-tailed test, the critical Z-value is ±1.96. Make a Decision: If Z > 1.96, reject the null hypothesis. Since 2.53 > 1.96, you reject the null hypothesis. Conclusion: The Z-test shows that the enzyme activity after drug administration is significantly different from the known average of 50 U/ mL. This suggests that the drug likely has an effect on enzyme activity. This approach is useful in biochemistry when comparing enzyme activities, concentrations of biomolecules, or other measurable parameters to known standards

17. ANOVA ANOVA (Analysis of Variance) is a statistical method used to compare the means of three or more groups to determine if there is a statistically significant difference between them. ANOVA is particularly useful when you have multiple groups and want to test whether their means are all equal or if at least one group differs significantly from the others. ANOVA can be used for both qualitative and quantitative data.  

18. How ANOVA Works: 1. Null Hypothesis (H₀): Assumes that all group means are equal. 2. Alternative Hypothesis (H₁): Assumes that at least one group mean is different. 3. F-Statistic : ANOVA calculates an F-statistic, which is a ratio of the variance between group means to the variance within the groups. A larger F-statistic indicates a greater difference between group means relative to the variance within groups. 4. p-value : The F-statistic is used to determine the p-value, which helps decide whether to reject the null hypothesis. A low p-value (typically < 0.05) suggests that there is a statistically significant difference between the group means.

19. Correlation To find out linear relation between two variables eg height and weight, temp and pulse etc. In order to find out whether there is significant association or not between two variables ( we may call them x and y) we calculate Co-efficient of correlation, represented by “ r xy ”.

20. Suppose, we have two variables x and y and we have individuals who have each one reading of x and one reading of y. the correlation coefficient is given by the formula

21. Correlation analysis Correlation coefficient (r), which ranges from -1.0 to +1.0 1. + ve value = two variable go together in same direction E. g.= age and atherosclerosis 2. - ve value =increase in value of one variable associated with decrease in value of another variable E. g.=age and quick reflexes 3. 0 value = no linear correlation between 2 variables E. g.=height and school grade of children

22. 1. Pearson Correlation Type : Parametric Description : Measures the linear relationship between two continuous variables. It assumes that the data is normally distributed and that the relationship between the variables is linear. Use : The Pearson correlation coefficient (r) ranges from -1 to 1, where: r = 1 indicates a perfect positive linear relationship. r= −1 indicates a perfect negative linear relationship. r=0 indicates no linear relationship. Test : Pearson's correlation test is used to determine whether there is a significant linear relationship between two continuous variables

23. Spearman's Rank Correlation Type : Non-parametric Description : Measures the strength and direction of the relationship between two ranked variables. It does not assume a linear relationship or normally distributed data. Use : The Spearman correlation coefficient ρ or r s ranges from -1 to 1, similar to Pearson, but it is based on the ranks of the data rather than the actual values. Test : Spearman's rank correlation test is used to assess the association between two variables when the data is ordinal, not normally distributed .

24. Regression To know in an individual case the value of one variable, knowing the value of other. We calculate regression coefficient of one measurement to the another. We denote independent variable as x and dependent variable as y.

25. The value of b is called regression coefficient of y upon x

26. Similarly, we can obtain regression of x upon y. Where b 1 is regression coefficient of x upon y. The function of regression is to provide means of estimating the value of one variable from the another.

27. Chi-square Test To determine if there is a significant association between two categorical variables. To find a relationship between a treatment type and the presence or absence of a certain disease. Example : Determine if there is an association between receiving a COVID-19 vaccine and health outcomes (healthy vs. non-healthy) among a sample of individuals. Data Collection Vaccination Status Healthy Non-Healthy Total Vaccinated 150 30 180 Unvaccinated 80 40 120 Total 230 70 300

28. Procedure: 1. Formulate Hypothesis Null Hypothesis (H ) and alternate hypothesis (H 1 )

29. Null Hypothesis (H ) : There is no association between vaccination status and health outcome. Alternative Hypothesis (H 1 ) : There is an association between vaccination status and health outcome. Step 1: Formulate Hypotheses

30. Step 2: Calculate Expected Frequencies Data Collection Vaccination Status Healthy Non-Healthy Total Vaccinated 150 30 180 Unvaccinated 80 40 120 Total 230 70 300

31. Step 3: Compute the Chi-Square Statistic Healthy Non-Healthy Vaccination Status O ij E ij O ij E ij Total Vaccinated 150 138 30 42 180 Unvaccinated 80 92 40 28 120 Total 230 70 300

32. Step 4: Determine Degrees of Freedom

33. Check the published probability table, On referring to Table, with 1 degree of freedom, the value of for a probability of 0.05 is 3.84 Since the calculated chi-square statistic (11.18) is greater than the critical value (3.841), we reject the null hypothesis . We accept the alternate Hypothesis . Step 5: Compare the chi-square statistic with the critical value Χ 2 Χ 2 Chi-square Table

34. Interpretation Alternate hypothesis: there is a statistically significant association between vaccination status and health outcome. This suggests that receiving the COVID-19 vaccine is associated with a higher likelihood of being healthy compared to those who are unvaccinated .

35. The Kruskal-Wallis test The Kruskal-Wallis test is a non-parametric statistical test used to determine if there are statistically significant differences between the medians of three or more independent groups. It is the non-parametric alternative to the One-Way ANOVA and is particularly useful when the assumptions of ANOVA, such as normality and homogeneity of variance, are not met.   T his can be used to compare three or more independent groups or samples.

36.   Hypotheses : Null Hypothesis (H₀): The medians of the different groups are equal. Alternative Hypothesis (H₁): At least one group has a median different from the others. How the Kruskal-Wallis Test Works:  1 . Ranking the Data : - Combine all the data from the groups into a single dataset. - Rank the data from the lowest to the highest, assigning ranks to the data points. If there are ties, assign the average rank to the tied values.  

37. 2 . Calculating the Test Statistic : - The test statistic for the Kruskal-Wallis test is denoted as (H). - H is calculated using the sum of ranks for each group, the number of observations in each group, and the total number of observations across all groups.   3. Determining Significance : - The H statistic is compared to a chi-square distribution with ( k-1 ) degrees of freedom, where k is the number of groups. - A p-value is obtained, and if this p-value is less than a predefined significance level (e.g., 0.05), the null hypothesis is rejected, indicating that there is a significant difference in the medians of the groups.

38. Steps in the Kruskal-Wallis Test:   1. Combine and Rank Data : - Example: Suppose you are comparing the effectiveness of three different diets on weight loss in three groups of participants: - Group A (Diet A): 5, 7, 6 - Group B (Diet B): 8, 9, 7 - Group C (Diet C): 4, 3, 5   - Combine the data: [5, 7, 6, 8, 9, 7, 4, 3, 5] - Rank the data: [3, 4, 5, 5, 5, 6, 7, 7, 8, 9]   2. Calculate the Test Statistic (H): - Calculate the sum of ranks for each group. - Compute the ( H ) statistic using the Kruskal-Wallis formula.  

39. 3. Compare ( H ) to the Chi-Square Distribution : - Use the calculated ( H ) value and compare it to the critical value from the chi-square distribution table with ( k-1 ) degrees of freedom. - Determine the p-value.   4 . Interpret the Results : - If the p-value is less than the significance level, reject the null hypothesis. - Conclusion: There is a statistically significant difference in the median weight loss among the different diet groups.

40. Wilcoxon signed-rank test The Wilcoxon signed-rank test is a non-parametric statistical test used to compare two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ. It's a popular alternative to the paired t-test when the data cannot be assumed to be normally distributed.

41. How It Works: Calculate Differences : For each pair of observations, calculate the difference between the two related samples. Rank the Differences : Rank the absolute values of these differences, ignoring any differences that are zero. Assign Signs to the Ranks : Assign the original signs (+ or -) of the differences to the ranks.

42. 4) Sum the Ranks : Sum the ranks separately for the positive and negative differences. 5) Test Statistic : The test statistic W is the smaller of the absolute values of these two sums (positive and negative ranks). 6) Compare to Critical Value or Compute p-value : This test statistic is then compared to a critical value from the Wilcoxon distribution table, or a p-value is computed to determine the significance of the test.

43. Significance Assessing Changes : The Wilcoxon signed-rank test is used to assess whether there is a statistically significant change in a population’s median between two conditions. For example, it can determine if a treatment has a significant effect on a group of patients when comparing pre-treatment and post-treatment scores. When to Use : This test is particularly useful when the assumptions of the paired t-test are not met, such as when the data is ordinal, not normally distributed, or when dealing with outliers that could affect the results of a parametric test. Interpreting Results : A significant result (p-value < 0.05, for example) suggests that there is a difference in the median ranks of the two samples, implying that the treatment or intervention had an effect.

44. Receiver Operating Characteristic  ( ROC)/   ROC curve is a  graphical  plot which illustrates the performance of a  binary classifier  system. It is created by plotting the fraction of  true positives / sensitivity(Y-axis)   vs. the fraction of  false positives / (1-specificity)(X-axis), at various threshold settings. In general, if both values are known, the ROC curve can be drawn.

45. Types - ROC curves representing excellent, good, and worthless tests plotted on the same graph. The accuracy of the test depends on how well the test separates the group being tested into those with and without the disease in question. Accuracy is measured by the area under the ROC curve   An area of 1 represents a perfect test; an area of .5 represents a worthless test. A rough guide for classifying the accuracy of a diagnostic test is the traditional academic point system: .90-1 = excellent (A) .80-.90 = good (B) .70-.80 = fair (C) .60-.70 = poor (D) .50-.60 = fail (F)

46. Sensitivity and Specificity

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