Statistical Analysis

Purpose of Statistical Analysis Determines whether the results found in an experiment are meaningful. Answers the question: –Does the experiment prove that the research hypothesis is true or false?

Example Study A researcher performs a clinical trial (randomized, controlled, double-blind experimental study) to determine whether a new medication is effective in lowering LDL cholesterol levels in patients with hypercholesterolemia. The researcher hypothesizes that the medication will be more effective in lowering LDL cholesterol levels in patients with hypercholesterolemia than a placebo.

First Step? Where should this researcher begin analysis of the data collected?

First: Identify the Hypothesis When statistically analyzing experiments, it is necessary to set up two hypotheses. The first hypothesis is called the null hypothesis. The second hypothesis is called the alternative hypothesis. Note: The alternative hypothesis should be set up before beginning the experiment.

Hypotheses Null Hypothesis (H 0 ): The starting point in scientific research where the experimenter assumes there is no effect of the treatment or no relationship between two variables. Example: The mean of group one is equal to the mean of group two. Alternative Hypothesis (H a ): (Also called research hypothesis) What the experimenter thinks may be true before beginning the experiment. Two types of alternative hypotheses: Directional and Nondirectional

Alternative Hypothesis Directional Alternative Hypothesis: Hypothesis predicting that the mean ( ) of one group will be more or less than the mean of the other group. Example: Nondirectional Alternative Hypothesis: Hypothesis predicting that the mean of one group is not equal to the mean of the other group(without specifying whether it will be more or less). Example:

Identify the Hypotheses for the LDL Medication Experiment Null Hypothesis: There is no difference in LDL cholesterol levels in patients given the medication than in patients given the placebo. Alternative Hypothesis: The LDL cholesterol levels will be lower in the patients given the medication than the patients given the placebo. –This is a Directional Alternative Hypothesis

Second: Determine Typical Data Values Determine what values are typical, or normal, for the data collected. Calculate the mean, variance, and standard deviation.

Mean Mean ( ): The arithmetic average.

Example Calculation for the LDL Medication Experiment

Next Step? Now that the mean has been calculated, it is important to determine how spread out the data is from the mean. –Two measures of spread are variance and standard deviation.

Variance Variance: The measure of the spread of the data about the mean. Variance is referred to as the average deviation of the data points from the mean. The more spread out the data points are, the larger the variance will be. Step One: Calculate the deviation (or difference of the data point from the mean) for each data point. Step Two: Square these deviations to ensure that all values are positive. Step Three: Calculate the sum of all of these squared deviations. Step Four: Divide by the number of data points in the data set minus one.

Calculating the Variation

Example Calculation for the LDL Medication Experiment

Standard Deviation Standard Deviation: The measure of the spread of the data about the mean, which is the square root of the variance. –When calculating the variance, you must square each difference in order to obtain all positive numbers. This results in large numbers. –The standard deviation is the square root of the variance, resulting in a number representative of the data set because it is in the same scale and same unit of measure as the original data points. This provides a more accurate measure of the spread than the variance. Step One: Calculate the variance for the data set. Step Two: Take the square root of the variance.

Calculating the Standard Deviation

Example Calculation for the LDL Medication Experiment

Importance of Calculating the Standard Deviation The standard deviation summarizes how close the data points are to the mean. We can say that the data set is normally distributed if: –Approximately 68% of the data falls within one standard deviation of the mean; –Approximately 95% of the data falls within two standard deviations of the mean; and –Over 99% of the data falls within three standard deviations of the mean.

What Do The Results Show? The results show that the mean of the LDL levels of the subjects given the experimental medication were lower than the LDL levels of the subjects given the placebo. How can the researcher determine whether the difference between the experimental group and control group in lowering blood LDL levels was actually due to the medication or due to chance? –i.e., Were the results statistically significant?

A type of statistical analysis called a: t-test

t-tests t-test: Type of statistical calculation used to determine whether the differences between the means of two samples are statistically significant. –Two main types of t-tests are commonly used to analyze biomedical data: Student’s t-test (i.e., independent t-test) Paired t-test (i.e., dependent t-test)

Student’s t-test Student’s t-test: Used to determine whether the difference between the means of two independent groups (both which are being tested for the same dependent variable) is statistically significant. Example: Study set up where one group is given the experimental treatment and another group is given the placebo.

There are three variations of the same formula for the student’s t-test: –The first variation should be used when the sample sizes are unequal AND either one or both samples are small (n<30). –The second variation should be used when the sample sizes are equal (regardless of size). –The third variation should be used when sample sizes are unequal AND both sample sizes are large (n>30).

Paired t-test Paired t-test: Used to determine whether the difference between the means of two groups (each containing the same participants and being tested at two different points) is statistically significant. Example: Study set up where the same group of participants is followed before and after an experimental treatment.

Formula for Paired t-test

Now Back to the LDL Medication Experiment Determine which type of t-test (student’s t- test or paired t-test) is most appropriate for our LDL levels experiment. –Because the two groups being tested are independent of each other (participants in the experimental group and the control group are different), the student’s t-test is the appropriate test to use.

Which variation of the formula should we use? –Both the experimental group and control group contain four participants. Because the sample sizes are equal, the second variation should be used.

Calculations:

You’re Not Done Yet What you just calculated is called the t value. Next you will use a t-table in order to determine whether your t value is statistically significant.

T-Table

Calculate the degrees of freedom for the experiment. How to Use a T-Table Step 1

Degrees of Freedom Degrees of Freedom: A measure of certainty that the sample populations are representative of the population being studied. –To calculate the degrees of freedom for a sample: Total # data points in sample(s) - # populations being sampled

Formulas:

Calculating the Degrees of Freedom for the LDL Medication Experiment

How to Use a T-Table Step 2 Find the row corresponding with the appropriate degrees of freedom for your experiment.

How to Use a T-Table Step 3 Determine whether the t value exceeds any of the critical values in the corresponding row. –If the t value exceeds any of the critical values in the row, the alternative hypothesis can be accepted. This means that the results ARE STATISTICALLY SIGNIFICANT. –If the t value is smaller than all of the critical values in the row, the alternative hypothesis can be rejected and the null hypothesis can be accepted. This means that the results are NOT STATISTICALLY SIGNIFICANT.

How to Use a T-Table Step 4 If the t value exceeds any of the critical values in the row, you need to: 1)Follow the corresponding row over to the right until you locate the column with the critical value that is just slightly smaller than the t value. 2)Follow this column to the top of the table and determine its corresponding p value. 3)Determine which p value to use, the p value corresponding with a one-tailed test for significance versus the p value corresponding with a two-tailed test for significance.

One-Tailed vs. Two-Tailed Test for Significance In order to determine whether you completed a one-tailed or two-tailed test for significance, look back to your alternative hypothesis. –If the alternative hypothesis was directional, you have completed a one-tailed test for significance. –If the alternative hypothesis was nondirectional, you have completed a two- tailed test for significance.

One-Tailed or Two-Tailed Test for Significance? The Alternative Hypothesis was as follows: –The researcher hypothesizes that the medication will be more effective in lowering LDL cholesterol levels in patients with hypercholesterolemia than the placebo. Since this is a directional alternative hypothesis, you have completed a one- tailed test for significance.

What Does the p Value Mean? The p values indicate the probability that the difference between the means of the two samples are due only to chance. –If the p value ≤ 0.01, the results are VERY SIGNIFICANT. The probability that the difference is due to chance is less than or equal to1%. –If the p value ≤ 0.05, the results are SIGNIFICANT. The probability that the difference is due to chance is less than or equal to 5%. –If the p value > 0.05, the results are NOT SIGNIFICANT. The probability that the difference is due to chance is greater than 5%.

Using the T-Table for the LDL Medication Experiment

Putting It All Together The group given the new medication had a mean of and a standard deviation of The group given the placebo had a mean of and a standard deviation of The t value for this study was 0.89, with a p value greater than This means that the results are NOT statistically significant. –This means that the researcher can reject the alternative hypothesis and accept the null hypothesis that the new medication did NOT lower patients’ LDL levels more than the placebo.