Biostatistics 6030 Review on other concepts and applications Descriptive statistics measures of central tendency Median How to calculate Order the observations from smallest to largest The sample median is The observation if The average of the even and is odd 1 observations if is Ordered array of ages of 189 subjects N 189 Order the data from smallest to largest Sample median th 95th one Counting from the smallest up to the 95th value we see that it is 54 Thus the median age of the 189 subjects is 54 years Descriptive statistics measures of central tendency The Mode A value which occurs most frequently in a set of values If all the values are different there is no mode A set of values may have more than one mode Ordered array of ages of 189 subjects A count of the ages reveals that the age 53 occurs most frequently 17 times The mode for this population of ages is 53 Descriptive statistics measures of central tendency Find the mode A laboratory with 10 employees whose ages are 20 21 20 20 34 22 24 27 27 and 27 A sample consisting of the values 10 21 33 53 and 54 May be used also for describing qualitative data The patients seen in a mental health clinic during a given year received one of the following diagnoses mental retardation organic brain syndrome psychosis neurosis and personality disorder The diagnosis occurring most frequently in the group of patients would be called the modal diagnosis Measures of Dispersion Percentiles Quantiles The percentile is the value such that percent of the sample points are less than or equal to The median is the percentile The The largest sample point if is not an integer where is the largest integer less than Tip round it up if is not a integer The average of the and largest observations if is an integer Tip average it up if is a integer Measures of Dispersion Example 1 Calculate 90th percentile of the following data 10 9 8 7 6 5 4 3 2 1 Example 2 Calculate 90th percentile of the following data 5 4 3 2 1 Screening tests Disease Test Present Absent Total Total number of subjects with disease and with a positive test number of subjects without disease and with a negative test number of subjects with a positive test BUT without disease FALSE POSITIVE among the healthy subjects number of subjects with a negative test BUT with disease FALSE NEGATIVE among the disease subjects Disease Test Present Absent Total Total Sensitivity probability of a positive test symptom given the presence of the disease Specificity probability of a negative test no symptom given the absence of the disease Predictive value positive PV probability of presence of disease given a positive test Predictive value negative PV probability of absence of disease given a negative test Binomial distribution Formal definition The distribution of the number of successes in statistically independent trials where the probability of success on each trial is is known as the binomial distribution and has a probability mass function given by Binomial distribution Example What is the probability of obtaining 2 boys out of 5 children if the probability of a boy is 51 at each birth and the sexes of successive children are considered independent random variables Solution A binomial distribution with Using binomial table What if the probability of success on an individual trial is greater than 5 Since The probability of successes with a success probability of is the probability of successes with a success probability of where Total probability rule Since and Hence To generalize let 1 2 be a set of mutually exclusive and exhaustive events a weighted average of the conditional probabilities of Total probability rule Example Let mammogram breast cancer Suppose that among 100 000 women with negative mammograms 20 will be diagnosed with breast cancer within 2 years or Pr whereas 1 woman in 10 with positive mammograms will be diagnosed with breast cancer within 2 years or Pr Suppose that 7 of the general population of women will have a positive mammogram Pr Q What is the probability of developing breast cancer over the next 2 years among women in the general population Total probability rule Estimation of the Mean of a Distribution Let be a random sample drawn from some population with mean Then for the sample mean Therefore is an unbiased estimator of Def We refer to an estimator of a parameter as An estimator of a parameter is unbiased if This means that the average value of over a large number of repeated samples of size is Standard Error of the Mean Let be a random sample from a population with underlying mean and variance The set of sample means in repeated random samples of size from this population has variance The standard deviation of this set of sample means is thus and is referred to as the standard error of the mean or the standard error Standard Error of the Mean In practice the population variance is rarely known We will then use the sample variance to estimate The standard error of the mean sem or the standard error se is given by theoretically and is estimated by in practice The standard error represents the estimated standard deviation for a set of sample means from repeated samples of size n drawn from a population with underlying variance Interval Estimation The interval is referred to as a confidence interval for Confidence Interval for the Mean of a Normal Distribution A CI for the mean of a normal distribution with unknown variance is given by or How to interpret a 95 CI Over a large number of hypothetical samples of size 10 95 of such intervals contain the parameter Any one interval from a particular sample may or may not contain the parameter We cannot say there is a 95 chance that the parameter will fall within a particular 95 CI Formal interpretation Over the collection of all 95 CIs that could be constructed from repeated random samples of size n 95 will contain the parameter Estimation for the Binomial Distribution Point Estimation of the Binomial Parameter Let be a binomial random variable with parameters and An unbiased estimator of is given by the sample proportion of events Its standard error SE is given exactly by and is estimated by The best estimate of the prevalence rate of malignant melanoma among 45 to 54 year old Its estimated standard women is error SE is Estimation for the Binomial Distribution Normal Theory Method for Obtaining a CI for the Binomial Parameter Wald Method An approximate CI for the binomial parameter based on the normal approximation to the