Statistics – 1

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57 Statistics – 1

Chittaranjan Andrade

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1. ANOVA is used to compare two groups.
2. Scales with multiple answers increase reliability in measuring attitude.
3. The chi-squared test of a 5 by 3 contingency table will have 15 degrees of freedom.
4. The chi-squared test could be used in a study of response to a new antidepressant in four different hospital populations.
5. Correlation is used to compare two groups.
6. Regarding correlation coefficient ‘r’, a value of +1 or −1, implies a cause–effect relationship.
7. Knowing the outcome status in a cohort study may lead to information bias.
8. A type II error occurs when a real difference is missed.
9. Observer-rated scales are prone to ‘halo effect’.
10. The least squares method is used for dichotomous variables.
11. Logistic regression is done by the least squares method.
12. Means taken from large samples of a skewed distribution approximate to a normal distribution.
13. T² test is used for non-parametric variables.
14. Normal distribution is a requirement for parametric testing.
15. The power of a test varies with the size of the sample.
16. Volunteer bias would deter healthy people from responding to postal questionnaires.
17. R² usually increases if a further independent variable is added.
18. Operational definition of concepts used in rating scales increases reliability.
19. Newspaper advertising can be used for random sampling.
20. Psychiatric screening instruments used in a general hospital are usually used to detect uncommon syndromes.
21. In a trial of two antidepressants, failure to show a difference in outcome may be due to there being less than 5% of subjects with previous episodes of depression.
22. A result statistically significant at a level of 5% implies that the correlation coefficient is 0.95.
23. A sensitive test has a low number of false positives.
24. Split-half reliability measures the validity of a test.
25. In an experiment to determine whether drug therapy or hypnosis is better as a treatment to stop smoking, the results are measured by the number of cigarettes smoked after treatment. In this study, hypnosis is a dependent variable.

ANSWERS

1. ANOVA is used to compare two groups.

True: ANOVA (analysis of variance) is a test used to compare means across two or more groups. Thus, we can use ANOVA to compare the mean IQ scores of men and women in a sample; or, we can use ANOVA to compare the mean baseline depression scores of patients randomized to experimental antidepressant drug (active control) and placebo (inactive control) groups. When there are only two groups, we often use the independent sample t-test because it is easier to compute.

2. Scales with multiple answers increase reliability in measuring attitude.

False: In general, the larger the number of categories in a variable, the more difficult it becomes for a test to yield significant results. Therefore, if a scale has multiple answers for each item, it becomes more difficult to achieve high reliability for a sample of given size.

3. The chi-squared test of a 5 by 3 contingency table will have 15 degrees of freedom.

False: The degrees of freedom for a chi-squared contingency table is computed by the formula (Rows–1) × (Columns–1). In this case, it will be (5–1)(3–1); that is, 4×2; that is, 8.

4. The chi-squared test could be used in a study of response to a new antidepressant in four different hospital populations.

True: True, if patients are classified as responders and non-responders. Then, we would compare the proportion of responders to non-responders in the four different hospital groups.

False, if response is assessed as a continuous variable, such as using the Hamilton Depression Rating Scale. One-way ANOVA would then be a more appropriate test.

Given that this question does not specify whether response is measured as a continuous variable or as a binary outcome the answer is true – chi-squared could be used.

5. Correlation is used to compare two groups.

False: Correlation is used to measure the strength of the relationship between two (usually continuous) variables. For example, we can use correlation to measure the strength of the association between IQ scores and marks obtained in an examination, or between serum antidepressant levels and percentage improvement in depression ratings. When the distribution of the variables is normal we use Pearson’s product moment correlation. When the distribution is significantly non-normal the variables are ranked and a rank-order (Spearman’s) correlation is performed.

6. Regarding correlation coefficient ‘r’, a value of +1 or −1 implies a cause–effect relationship.

False: Values of +1 or −1 indicate a perfect correlation, i.e. as the value of one variable changes by a certain degree, the value of the other variable changes by an exactly corresponding degree (though not necessarily the same in absolute value). Correlation measures only the strength of association between two variables; it never implies causality. Consider a hypothetical situation. We wish to determine the correlation between the number of eyeballs and the number of toes in different households in a residential area. We find that one household has 2 eyeballs and 10 toes altogether; in another household, there are 8 eyeballs and 40 toes, and so on. Assuming that nobody is handicapped or otherwise abnormal there will be a perfect correlation between the number of eyeballs and the number of toes. But, neither causes the other!

7. Knowing the outcome status in a cohort study may lead to information bias.

False: In cohort studies we try to measure the outcome status. Knowing the exposure status in a cohort study, or the outcome status in a case-control study, would lead to information bias.

8. A type II error occurs when a real difference is missed.

True: A type II (false negative) error is said to occur when the null hypothesis is mistakenly accepted, i.e. when a significant relationship between variables remains unrecognized. Type II errors occur when the sample size is small, when the sampling is biased, and when heterogeneity in the sample is large (and hence when the standard deviations are large).

The probability of a type II error is known as beta. The power of a test is a measure of how likely the test is to avoid a type II error.

9. Observer-rated scales are prone to ‘halo effect’.

True: Halo effect is when the observer is biased by one positive aspect of the subject. For example, good looking candidates are more likely to pass viva examinations (Fear 2004, p. 32).

10. The least squares method is used for dichotomous variables.

False: The least squares method is used in regression to examine the relationship between continuous variables.

11. Logistic regression is done by the least squares method.

False: It is linear regression which employs the least squares method. The least squares method refers to the technique by which linear regression lines are fitted to a scatter plot of individual data points. The vertical distances between the individual data points and the line are calculated, squared and added together. The line of best fit is that line which has the smallest ‘sum of squares’ (Puri & Tyrer 1998, p. 278).

12. Means taken from large samples of a skewed distribution approximate to a normal distribution.

False: If the population distribution is skewed, this will be reflected in its standard deviation. A random sample drawn from such a sample will also be skewed, and this will also be reflected in the standard deviation of the sample. As the standard error of the mean (which describes the distribution of means of samples drawn from the population) is computed by dividing the standard deviation by the square root of the sample size, it follows that the standard error will also reflect the skewness. Hence, the means will NOT approximate to a normal distribution.

13. T² test is used for non-parametric variables.

False: The Hotelling’s T² test is used in multivariate analysis of variance (MANOVA), and is applied to continuous variables. NB: This is different from the student t-test.

14. Normal distribution is a requirement for parametric testing.

True: Tests such as the t-test, ANOVA, ANCOVA, RMANOVA, MANOVA and Pearson’s correlation are parametric procedures. They test dependent variables which are continuous in nature, i.e. operationalized along interval or ratio scales. For parametric testing, the dependent variable should be normally distributed. Incidentally, this requirement notwithstanding, parametric tests are generally robust to violations of normality, unless the skewness or kurtosis is marked.

If distributions are significantly non-normal, non-parametric alternatives, e.g. Wilcoxon test, Mann–Whitney test, Kruskall–Wallis test and Spearman’s correlation procedure are used. These tests are conducted on ranked data.

15. The power of a test varies with the size of the sample.

True: A larger sample is associated with greater statistical power.

16. Volunteer bias would deter healthy people from responding to postal questionnaires.

True: Persons who suffer from the problem addressed in the questionnaire would be more interested in the subject than healthy persons, and would therefore be more likely to take the trouble to respond to a postal questionnaire.

17. R² usually increases if a further independent variable is added.

True: The extent of increase in R² depends on the extent to which the new independent variable correlates with the dependent variable and explains an additional proportion of the variance in the dependent variable.

Here is the explanatory note. In multiple regression, R is the correlation between the actual value of the dependent variable in an index case and the value of the dependent variable predicted by the regression equation for that case. Expressed otherwise, R is the multiple correlation coefficient, much as r is the correlation coefficient between two continuous variables. Just as r² (the square of the correlation coefficient) is the proportion of the dependent variable explained by the independent variable, so too is R² the proportion of the dependent variable explained by the different independent variables in the multiple regression equation.

18. Operational definition of concepts used in rating scales increases reliability.

True: Operationalization of a variable is the procedure whereby a variable is defined in a way that permits its accurate (or at least standardized) measurement.

For example, the question ‘How did you sleep last night?’ may have three choices as answers: Very poorly, Poorly, and Well. Different patients may make different choices despite having slept for the same period. This can be more reliably assessed if the choices are operationalized as follows:

• Well: Had the necessary quota of sleep, felt refreshed on awaking

• Poorly: Experienced disturbed sleep, lost up to 2 hours of sleep

• Very poorly: Lost more than 2 hours of sleep.

19. Newspaper advertising can be used for random sampling.

False: Subjects recruited through newspaper advertisements are those who have direct or indirect access to a newspaper, those who are interested in the subject of the study, those who have the personality characteristics which make them more likely to respond or volunteer, those who may be impoverished and thereby more interested in the inconvenience reimbursements offered for completing the study, etc. Such subjects, by virtue of these biases, are not representative of the population. Newspaper advertisements for sampling are useful when recruitment for the study is slow or difficult.

20. Psychiatric screening instruments used in a general hospital are usually used to detect uncommon syndromes.

False: Screening instruments are used to quickly and conveniently identify patients who may have a disorder. Thus, screening instruments can be applied in group settings, or to large samples of patients. Thereby, screening is a time-cost-energy effective substitute for more accurate but more expensive or time-consuming procedures, and can be used to detect any syndrome, common or uncommon.

21. In a trial of two antidepressants, failure to show a difference in outcome may be due to there being less than 5% of subjects with previous episodes of depression.

False: Insufficient statistical power or a genuine lack of any difference between the two drugs would be the only reasons. Insufficient statistical power could be due to:

1. Insufficient sample size

2. The use of a categorical outcome variable (which would require a larger sample size) instead of a continuous outcome variable

3. Sample heterogeneity (which would increase the standard deviation and hence necessitate a large sample size)

4. High similarity of antidepressant effect of the two drugs (which would require a larger sample size to detect the small difference).

22. A result statistically significant at a level of 5% implies that the correlation coefficient is 0.95.

False: Statistical significance may relate to means, proportions, or other matters, and not necessarily to the correlation coefficient. In the case of the correlation coefficient, a very large correlation coefficient may not be significant if the sample size is small. And, a very small correlation coefficient may well be significant if the sample size is very large.

23. A sensitive test has a low number of false positives.

False: It has few false negatives.

Sensitivity is the ability of the test to identify the presence of disease.

Sensitivity = true positives/total cases.

Specificity is the ability of the test to identify the absence of disease.

Specificity = true negatives/total non-cases.

Positive predictive value is the proportion of cases that actually have the disease out of the total number of cases predicted by the test to have the disease.

Negative predictive value is the proportion of cases that actually are disease free out of the total number of cases predicted by the test to be disease free.

Efficiency is the ability of the test to correctly identify presence and absence of disease, i.e. the proportion of cases correctly classified.

Note that sensitivity and specificity are often inversely related. A sensitive test often has low specificity, and vice versa.

24. Split-half reliability measures the validity of a test.

False: Split-half reliability is one way to measure the internal consistency of a test which purports to measure a single construct. It is a test of reliability. In split-half reliability, the items in the test are split into two equal halves and the scores for the two halves are correlated. If the correlation is high, the test is internally consistent, i.e. the halves of the test are likely measuring the same thing. Cronbach’s alpha is a special way of measuring split-half reliability. It is the average correlation of all the possible ways in which the test items can be divided into two halves.

25. In an experiment to determine whether drug therapy or hypnosis is better as a treatment to stop smoking, the results are measured by the number of cigarettes smoked after treatment. In this study, hypnosis is a dependent variable.

False: Treatment (drug therapy vs. hypnosis) is the independent variable, and number of cigarettes smoked is the dependent variable.

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