Research Question Do nicotine-exposed rats have different X gene expression than control rats in ventral tegmental area? Design an experiment in which treatment rats (N>2) are exposed to nicotine and control rats (N>2) are exposed to saline. Collect RNA from VTA, convert to cDNA Determine the amount of X transcript in each individual. Perform a test of means considering the variability within each group.
Observed difference between groups May be due to Treatment Chance
Hypothesis Testing Null hypothesis: There is no difference between the means of the groups. Alternative hypothesis: Means of the groups are different.
Hypothesis testing You can not accept null hypothesis You can reject it You can support it
P-value The ‘P’ stands for probability, and measures how likely it is that any observed difference between groups is due to chance, alone.
P-value there is a significant difference between groups if the P value is small enough (e.g., <0.05). P value equals to the probability of type I error. Type I error: wrongly concluding that there is a difference between groups (false positive). Type II error: wrongly concluding that there is no difference between groups (false negative).
Multiple tests on the same data Expression data on multiple genes from the same individuals Subsets of genes are coregulated thus they are not independent. Such data requires multiple tests.
Why not do multiple t-tests? Or if you do, adjust the p-values Because it increases type I error: a study involving four treatments, there are six possible pairwise comparisons. If the chance of a type I error in one such comparison is 0.05, then the chance of not committing a type I error is 1 – 0.05 = 0.95. then the chance of not committing a type I error in any one of them is 0.95 6 = 0.74. Cumulative type I error = 1-0.74=0.26
Normal Distribution it is entirely defined by two quantities: its mean and its standard deviation (SD). The mean determines where the peak occurs and the SD determines the shape of the curve.
Rules of normal distribution 68.3% of the distribution falls within 1 SD of the mean (i.e. between mean – SD and mean + SD); 95.4% of the distribution falls between mean – 2 SD and mean + 2 SD; 99.7% of the distribution falls between mean – 3 SD and mean + 3 SD.
Most commonly used rule 95% of the distribution falls between mean – 1.96 SD and mean + 1.96 SD If the data are normally distributed, one can use a range (confidence interval) within which 95% of the data falls into.
A sample Samples vary Samples are collected in limited numbers They are representatives of a population. A sample: E.g., nicotine treated rat RNA
Sample means Consider all possible samples of fixed size (n) drawn from a population. Each of these samples has its own mean and these means will vary between samples. Each sample will have their own distribution, thus their own std.
Population mean The mean of all the sample means is equal to the population mean ( ). SD of the sample means measures the deviation of individual sample means from the population mean ( )
Standard error It reflects the effect of sample size, larger the SE, either the variation is high or sample size is small.
Confidence Intervals a confidence interval gives a range of values within which it is likely that the true population value lies. It is defined as follows: 95% confidence interval (sample mean – 1.96 SE) to (sample mean + 1.96 SE). a 99% confidence interval (calculated as mean ± 2.56 SE)
T-distribution The t-distribution is similar in shape to the Normal distribution, being symmetrical and unimodal, but is generally more spread out with longer tails. The exact shape depends on a quantity known as the ‘degrees of freedom’, which in this context is equal to the sample size minus 1.
One-sample t-test Null hypothesis: Sample mean does not differ from hypothesized mean, e.g., 0 (Ho: =0) A t-statistics (t) is calculated. t is the number of SEs that separate the sample mean from the hypothesized value. The associated P value is obtained by comparison with the t distribution. Larger the t-statistics, lower the probability of obtaining such a large value, thus p is smaller and more significant.
Paired t-test Used with paired data. Paired data arise in a number of different situations, a matched case–control study in which individual cases and controls are matched to each other, or A repeat measures study in which some measurement is made on the same set of individuals on more than one occasion
Two-sample t-test Comparison of two groups with unpaired data. E.g., comparison of individuals of treatment and those of control for a particular variable. Now there are two independent populations thus two STDs
Calculation of pooled STD The pooled SD for the difference in means is calculated as follows:
Calculation of pooled SE the combined SE gives more weight to the larger sample size (if sample sizes are unequal) because this is likely to be more reliable. The pooled SD for the difference in means is calculated as follows:
Two sample T-test Comparison of means of two groups based on a t-statistics and its student’s t-distribution. dividing the difference between the sample means by the standard error of the difference.
T-statistic A P value may be obtained by comparison with the t distribution on n1 + n2 – 2 degrees of freedom. Again, the larger the t statistic, the smaller the P value will be.
Example X-gene exprs.TumorControl # of samples119117 Mean8195 Std1819
T-statistic t = (95-81)/2.41 = 14/2.41 = 5.81, with a corresponding P value less than 0.0001. Reject null hypothesis that states that sample means do not differ.
Analysis of Variance ANOVA A technique for analyzing the way in which the mean of a variable is affected by different types and combinations of factors. E.g., the effect of three different diets on total serum cholesterol
Sources of variation P value of 0.0039 means that at least two of the treatment groups are different.
Multiple Tests Post hoc comparisons between pairs of treatments. Overall type I error rate increases by increasing number of pairwise comparisons. One has to maintain the 0.05 type I error rate after all of the comparisons.
Bonferroni Adjustment 0.05/#of tests Too conservative
NonParametric methods Many statistical methods require assumptions. T-test requires samples are normally distributed. They require transformations Nonparametric methods require very little or no assumptions.
Total drug doses in patients with a 3 to 5 day stay in intensive care unit. Rank all observations in the increasing order regardless of groupings Use average rank if the values tie Add up the ranks Select the smaller value, calculate a p- value for it.
Correlation and Regression Correlation quantifies the strength of the relationship between two paired samples. Regression expresses the relationship in the form of an equation. Example: whether two genes, X and Y are coregulated, or the expression level of gene X can be predicted based on the expression level of gene Y.
Product moment correlation r lies between -1 and +1
Method of least squares The regression line is obtained using the method of least squares. Any line y = a + bx that we draw through the points gives a predicted or fitted value of y for each value of x in the dataset. For a particular value of x the vertical difference between the observed and the fitted value of y is known as the deviation or residual. The method least squares finds the values a and b that minimizes the sum of squares of all deviations.