Business Statistics Independent and Dependent Variable

Question: 1. A researcher claims that among people older than 70 years in a particular city a higher proportion of women use full dentures than men. Suppose that you set up a study to assess the researchers claim. a) What are the appropriate hypotheses? b) How you would set up the study? c) What variables would you collect what is the data type of each variable? d) How you would summarize each variable? e) What statistical test you would apply? f) How you would interpret the results of such a test? g) How would the size of your sample tend to affect the results? 2. You are designing a study to measure periodontal disease in a particular population. a) Define and describe inter-examiner variability and intra-examiner variability. b) What steps would you take to attempt to minimise inter-examiner variability in your study. c) Describe problems that may be encountered when attempting to measure the severity of periodontal disease. d) What would be the characteristics of an ideal dental disease index? Illustrate your answer with reference to well-known indices used in dentistry. 3. Explain with examples what is meant by null and alternative hypotheses and how they are used in statistical testing. Suppose that you conducted an independent samples t-test and the analysis resulted in a t value of 1.3 and a p value of 0.20. What do these values mean and what conclusions would you draw? (30 marks) What does a p value of 0.01 mean and what conclusions would you draw? What is the relationship between: (a) The confidence interval for the difference between the means of two groups and (b) The p value given by a t-test comparing these two groups? 4. What do you understand by the following statistical and epidemiological terms? Illustrate each with an example. a) Multistage sampling b Multiplication law of probability c) Posthoc test d) Sensitivity of a diagnostic test e) A double blind study f) Histogram 5. Explain briefly the principles behind, and the use and limitations of three of the following. Suggest at least one situation where each might be used when analysing dental data. (Equal marks for each sub-section) a) Receiver Operating Characteristic (ROC) curve b) Wilcoxon matched-pairs signed-ranks test c) Cohens kappa coefficient d) Spearman's rank correlation coefficient 6. Explain the principles behind linear regression. Describe a situation in dentistry or dental research where linear regression might be a useful technique. What assumptions are made when linear regression is used and for each assumption suggest a technique that might be employed (when data violates the assumption) to make use of linear regression a appropriate? Answer: 1. In order to study the use of full dentures in a particular city across their genders, the population of people over 70 years are chosen. In order to set up the study, the research null hypothesis and alternate hypotheses are as follows: H0 : There is no difference in the proportion of usage of dentures across genders H1 : proportion of women use more full dentures than the proportion of men The study would involve selection of the population that includes people over 70 years in a particular city. Both male and female people would be selected as a population of the study. The samples would be selected from this population randomly. The chosen samples would be studied to find out whether they use dentures or not. Statistical methods would be used to analyse the chosen samples. Proportion of males in the samples who use full denture and the proportions of females in the samples who use denture would be found out. Comparisons among the males and females of the chosen sample would be done and z test would be performed in order to test the hypotheses. The variables that would be selected for this study consist of males and females of the city above the age of 70 years. There would be two variables in this research; the males above 70 years who use full dentures and the females above 70 years who use dentures. Data would also be collected for the people who do not use full dentures. Proportions of males and females would be calculated from the collected data. Quantitative data would be collected for this study. Each of the variables would possess quantitative data. The variables would have discrete values and they will be primary data. The two variables consist of proportion of males and proportion of females above 70 years in the city who use full dentures. The number of males who use full dentures and number of males chosen in the sample would constitute the proportion of males above 70 years who use dentures and the number of females who use full dentures and number of females chosen in the sample would constitute the proportion of females above 70 years who use dentures. Two types of statistical tests would be used in this study. They are methods of frequency and z- test. The methods of frequency would be used to count the number of males and the number of females in the population of the city above 70 years who use full dentures. This method would help to find the proportion of males above 70 years who use full dentures and the proportion of females above 70 years who use full dentures. The method of z-test would be used to perform the hypotheses tests. Z-test would be used to test the difference in means among the proportions of two samples when their variances are known. This is because those samples are drawn from the population whose variances are known. The result would be interpreted on seeing the p value of the test. One tailed z-test would be performed in this study. On considering the level of significance as 0.05, the p value of the test would be compared with the level of significance as 0.05. If the p value of the test is, upper-tailed test is less than 0.05, the null hypothesis is rejected and if the p value of the upper-tailed test is more than 0.05, the null hypothesis is accepted. The size of the sample is very important in order to conduct a proper hypothesis test. More the size of the sample, more effective will be the test. There would be change in the size of confidence interval on change in the size of the sample. It would affect the result of the test, as the p value of the test would change due to the change in degrees of freedom. This, in turn would affect the rejection or acceptance of the null hypothesis. 2. In order to study the measure of periodontal disease in a particular population, the population must be divided in various classes in context of age groups. Every age group would form a class, which consist of persons having periodontal disease. Inter-examiner variability is defined as the variation of the sample across different classes. The interclass-examiner variability would examine the variability of the disease across different age groups of the population. The occurrence of the disease across the age groups would be analysed using inter-examiner variability. Intra-examiner variability is defined as the variability of the samples within their class. In context of periodontal disease, the samples were divided into different classes. The variation of the samples within their class would form intra-examination disease. Intra-examiner variability would measure the variation of periodontal disease within every age group of the samples. In order to minimise the inter-examiner variability in the study, the samples selected for the study should be such that the samples have similar number of periodontal disease in every age group. The count of the type of periodontal disease must be same across every age group. This would lead to the reduction in variability in periodontal disease across the age groups, thereby reducing inter-examiner variability. The total number of samples in each of the age groups must be same and this would result to less variability during inter-examiner. Many problems may be encountered to measure the severity of periodontal disease. This ranges from setting the level of severity and judging them while collecting data. The measure of degree of severity varies from person to person. Severity is a qualitative measure and it becomes difficult to set the range beyond which it can be depicted as severity. People who have been diagnosed with periodontal disease undergo treatment for this disease. This leads to decrease in the extent of the disease with time. When the researcher will go to collect the samples for the survey, he might find the degree of severity of the disease less. This would lead to improper collection of the data. Another problem that might be faced while collecting data is the problem of biasness. People may not be open to confess their periodontal disease to the researcher due to ethical issues. They may hesitate to give the correct response and the data would become bias and incorrect for the research. The characteristics of an ideal dental disease index are validity, reliability, clarity, simplicity and objectivity, quantifiabiltiy, sensitivity and acceptability. The well-known dental index is DMF index. This is an irreversible index and is applied for permanent teeth. This is the best known and most widely used dental index which judges a teeth according the severity of periodontal disease. The severity is judged based on decay, missing and filled status of the tooth. Another well-known index used in dentistry is the Graingers hierarchy index, which is an ordinal scale index with five zones of severity attack. There are various degrees of severity and the scale of the index is judged by the condition of the teeth and decay it had undergone. 3. Null hypothesis and alternate hypothesis are both parts of inferential statistics. These hypotheses are used to test any hypothesis test and draw inference from it. Null hypothesis is referred to as the general position or default position of any statement. This suggests that there is no relationship between two or more variables or there is no effect on a variable. Alternative hypothesis is referred to the statement which is hoped to be true against the null hypothesis. The symbol of alternative hypothesis is H1 while the symbol of null hypothesis is H0. An example of null hypothesis and alternative hypothesis will be shown while finding the effect of inflation on the food prices. In order to test this, the null hypothesis is: H0: inflation do not have any effect on the prices of foods of a country; and the alternative hypothesis is H1 : inflation effects the prices of food of a country. Alternative hypothesis can be either one sided alternative or two sided alternative. On analysing a certain analysis and performing an independent sample t-test, it was found that the t- value was 1.3 and the p value was 0.20. T-value is defined as the test statistic of t-test and it measures the difference between the hypothesis population parameter and the observed sample statistic in terms of standard error. The t-value is 1.3 can be interpreted that the difference between the observed sample statistic and the hypothesis population parameter in terms of standard error is 1.3. T-value is used to compare with the critical value of t-distribution considering (n 1) degrees of freedom. This gives an idea about the acceptance and rejection of null hypothesis. P-value is defined as the probability of finding the observed result when the null hypothesis is true. The standard p value is considered as 0.05. Here, p value is 0.20 can be interpreted that the null hypothesis will be accepted as the p-value of the test is greater than 0.05 and there is no difference between the means of the variables of the test. Both the confidence interval for the difference between the means and the p value of the given t-test helps to draw the inference of the hypothesis test. Confidence interval is used in case of classical approach to test the hypothesis whereas p value is used in case of probabilistic approach to test the hypothesis. The test statistic is used to check whether it lies within the confidence interval or not and draw the inference of the hypothesis accordingly. Whereas, p value of the test is compared with the standard p value; i.e. 0.05 to find if the null hypothesis is rejected or accepted. 4. Multistage sampling multistage sampling is defined as the method of sampling which involves sampling at various stages. This method of sampling uses smaller to smaller units of sampling at every stages thereby leading to refining of the samples according to much refined criteria at every successive stages. Multistage sampling is a complex form of cluster sampling as it involves dividing the population into clusters. Successive clusters are formed by choosing the samples randomly from within the chosen clusters. Multistage sampling is useful when it is essential to choose samples randomly from selected clusters. Multistage sampling has various advantages that includes efficiency of cost and speed of the survey and more accuracy over cluster sampling. However, testing of the samples selected by multistage sampling is difficult. An example of this sampling is given while conducting household survey in Australia. Australia had been divided into various metropolitan regions. It was th en divided into collection districts which was further divided into blocks. Multiplication law of probability the law of multiplication of probability includes the probability that both the events A and B had occurred is equal to the probability that event A had occurred multiplying with the probability that event B had occurred given event A had already occurred. An example of this rule of probability is that an urn contains 8 red balls and 5 white balls and two marbles are drawn from the urn without replacement. This can be solved on applying multiplication law of probability. Posthoc test this test is conducted in design of experiments which involves looking into the data after conducting the experiments. This test is done to confirm the occurrence of the differences between the groups. This test is run only after proving the that there exists overall significant differences between the means of the groups. This test helps to find the patters that were not specified earlier for the tests. An example of posthoc test can be shown after performing the t-test across the age groups of the students who did adventurous sports. Once the t-test had proved that there was difference of means between the age groups, post hoc test would be performed to confirm the occurrence of the difference between the groups. Sensitivity of a diagnostic test sensitivity of diagnostic test is used in medical diagnosis which tests the ability of the test to identify the disease and its cause correctly; i.e. true positive rate. An example of sensitivity of a diagnostic test is performing blood test on occurrence of diseases like HIV. The ability of the blood test to correctly identify HIV in the patients body is known as the sensitivity of a diagnostic test. A double blind study this is a study in which neither the participant nor the experimenter have any idea about the person who is receiving the particular treatment. A double blind study is done in order to prevent the biasness in the experiment. The respondent and the experimenter both are masked about the information of the experiment until an outcome of the trail is known. This helps to reduce the biasness of the experiment. An example of a double blind study can be given for the studies done to test any drug. The respondents and the experimenter are kept unknown from the experiment and they are provided with the drug unknowingly. The outcome is analysed and the effect of the drug is known. Histogram Histogram is a graphical representation of the numerical data. It is drawn for the continuous data and is represented by range of rectangles. The height of the rectangles is proportional to the frequency of the variable. An example of drawing histogram lies in representation of the distribution of rainfall at every dates of a chosen month at a particular place. The collected data is represented pictorially in form of histogram. 5. Receiver Operating Characteristic (ROC) curve This is a graphical plot that is used to illustrate the performance of a binary classifier system as there is a change in the threshold of the system. The principle of plotting this curve is by plotting the true positive rate against the false positive rate for various settings of threshold. Receiver Operating Characteristic (ROC) curve is used to evaluate the diagnostic performance of any test or to test the accuracy of a test to differentiate between the diseased cases and normal cases. The limitation of Receiver Operating Characteristic (ROC) curve that excessive extrapolation of the ROC curve in not desirable and it might be difficult to assign the confidence score to build ROC curve. Wilcoxon matched-pairs signed-ranks test this test is a non parametric statistical hypothesis test and is used to compare two related samples or repeated measure on a single sample in order to find if there is any difference in the population mean ranks. The limitation of this test is that this is a non parametric test and it cannot be used for parametric tests. Spearman's rank correlation coefficient this is used to find the correlation coefficient between two variables. The values of every variable are arranged in ranks and then correlation is done among the ranks of the variables. This test is used to identify the strength of relationship between the two variables. The limitation of this test is that the data must be ranked prior to performing the test. 6. The principle of linear regression provides an idea that there must be one dependent variable and one or more independent variable. The correlation coefficient is used to define the strength of association between the two variables while re square of the correlation coefficient defines the total variability in this regression. The value of correlation coefficient must lie within -1 to 1 while the value of square of correlation coefficient must lie between 0 and 1. The independent variables must not be correlated to one another in order to perform linear regression. In dental research, the decay of teeth is related to various factors like types of food consumed, care of teeth, any other diseases and regular checkups of teeth. Here decay of teeth is considered as dependent variable while the other factors are considered as independent variables. Decay of teeth is tested on the basis of the factors like types of food consumed, care of teeth, any other diseases and regular checkups of teeth. Regression analysis will be performed to find the degree of dependency of decay of teeth on the factors like types of food consumed, care of teeth, any other diseases and regular checkups of teeth. The assumptions that are made when linear regression is considered is that the variables are independent and they do not exhibit any co linearity among themselves. It is also assumed that they are homoscedasticity when these techniques are violated; linear regression method cannot be used. Various types of nonlinear techniques must be used in order to perform regression analysis on the variables. Otherwise, the non linearity must be removed from the data prior to performing linear regression tests.