Biostats Exam 3

general null hypothesis definition

a statement that the value of a population parameter is equal to some value

general alternative hypothesis definition

a statement that the parameter has a value that somehow differs from the null hypothesis

general null hypothesis symbolic notation

H0: mean = expected value

general alternative hypothesis symbolic notation

HA: mean =/ (does not equal) expected value

steps of a hypothesis test

1. establish null and alt. hypotheses

2. select a significance level

3. select an appropriate statistical test

4. collect sample and summarize the data into a test statistic

5. decide whether the result is statistically significant based on critical and/or p-value

alpha level definition

-aka significance level

-the probability value used as the cutoff for determining when the evidence is significant against the null hypothesis

-the probability of mistakenly rejecting the null hypothesis when it is true (type 1 error)

- = 1 - confidence levels

-commonly 0.01, 0.05, and 0.10

test statistic definition

-a value used in making a decision about the null hypothesis

-found by converting a sample statistic to a score with the assumption that the null hypothesis is true

-different for each statistical test used

-calculated by technology when running the particular statistical test

p-value definition and usage

-tells you how likely it is that the test statistic could have occurred under the null hypothesis

-the probability of getting a value of the test statistic that is at least as extreme as the actual calculated test statistic, assuming that the null hypothesis is true

-used to decide if a test is statistically significant

-p-value > alpha: fail to reject null hypothesis

-p-value < alpha: reject the null hypothesis

one-sample t-test in R

-t.test(x, mu, alternative)

-x = sample values

-mu = known population mean or value of comparison

-alternative = "less", "greater", or "two.sided"

independent t-test assumptions, usage, and distinguishing features

-compares the mean scores of 2 different GROUPS of people or conditions

-independent groups- the measurement of an individual in one group is unrelated to measurements in the second group

-observations are independent and randomly selected

-populations are normally distributed or n > 30

-test for normality using quantile-quantile plot of dependent variable

independent t-test in R

-t.test(x, y, alternative)

-x = dependent variable

-y = group

-alternative = "less", "greater", or "two.sided"

paired t-test assumptions, usage, and distinguishing features

-compares the mean scores for the same group of people on 2 different OCCASIONS

-simple random sampling is used

-the 2 groups of data are dependent

-the differences of individuals between 2 observations approximately follow a normal distribution or n > 30

-test for normality using quantile-quantile plot of difference between groups

paired t-test in R

-t.test(after, before, alternative, paired)

-after - dataset$after

-before - dataset$before

-alternative - "less", "greater", or "two.sided"

-paired = TRUE

generic independent t-test null and alternative hypotheses

-H0: mean(difference) = 0

-HA: mean(difference) < or > 0

generic paired t-test null and alternative hypotheses

-H0: mean(difference) = 0

-HA: mean(difference) < or > 0

what is the difference in using a comma or a tilda to separate the data sample inputs in R

-comma: x - y = ?

-tilda: alphabetical

chi-squared goodness of fit test variables, purpose, and requirements

-counts of categorical variables (frequencies)

-frequency tables (2 columns)

-used to test the hypothesis that an observed frequency distribution fits some expected distribution

-the sample data consists of frequency counts for each of the different categories

-the data are randomly selected

-for each category, the expected frequency is at least 5

chi-squared tests of independence variables, purpose, and requirements

-counts of categorical variables (frequencies)

-contingency tables (3+ columns) which consist of frequency counts of categorical data corresponding to 2 different variables

-test whether or not the row and column variables are independent

-the sample data are randomly selected

-the sample data are represented as frequency counts in a two-way table

-for every cell in the contingency table, the expected frequency (E) is at least 5

chi-squared distribution basic features

-continuous distribution

-minimum value of 0

-right-skewed

-shape determined by degrees of freedom (# of categories - 1)

-mean is the same as the degrees of freedom

generic chi-squared goodness of fit null and alternative hypotheses

-H0: the frequency counts agree with the expected distribution

-HA: the frequency counts do not agree with the expected distribution

generic chi-squared tests of independence null and alternative hypotheses

-H0: success or failure are independent of treatment type

-HA: success or failure are dependent on treatment type

chi-squared goodness of fit test in R

-chisq.test(x, p)

-x = a vector with the observed values

-p (optional) = a vector of probabilities, if the expectation is NOT that all probabilities are equal

making a contingency table in R

-(x, ncol)

-x = a vector with the observed values, entered one column at a time, starting with the left-most column

-ncol = number of columns

chi-squared tests of independence in R

-chisq.test(x)

-x = a matrix with the observed values

calculating expected values for a chi-squared test of independence

E = (row total * column total) / (grand total)

what are the main 2 problems of performing multiple t-tests?

-time consuming

-increase false positive probability

generic ANOVA null and alternative hypotheses

-H0: the means of all groups are equal

-HA: at least one group's mean is different

changes in variance effect on the F-statistic

-F = (variance between groups) / (variance within the groups)

-F = (n * s (2/x) / (s^2 pooled)

-greater variance between groups increases F

-greater variance within the groups decreases F

ANOVA assumptions

-random and independent observations

-assumption of normality or n > 30

-test normality using quantile-quantile plots

-assumption of homogeneity of variance

-test homogeneity of variance using Bartlett's test

Bartlett's test in R

-bartlett.test(y ~ x)

-y = the dependent variable (numbers)

-x = the independent variable (category names)

-P < 0.05 indicates that the groups do NOT have the same variance

ANOVA in R

-Bartlett's test

-aov(y ~ x)

-y = dependent variable (numbers)

-x = independent variable (category names)

-save ANOVA as an object

-summary(ANOVA object)

-Tukey's Honest Significant Different test

Tukey's Honest Significant Different Test in R (posthoc)

-TukeyHSD(ANOVA object)

Pearson correlation distinguishing features and assumptions

-provides information on the strength and direction (positive or negative) of a linear relationship between 2 variable

-no causation inferred

-effects on variables are symmetric

-data are independent and random

-variables are quantitative

-both variables are normally distributed

-correlation coefficient alone cannot tell us if results are significant

Linear regression distinguishing features and assumptions

-provides an explicit formula for a line that can be used to predict values of one variable based on the other

-provides a robust measure of "fit"

-only accounts for uncertainty in the dependent variable

-often interpreted as one variable causing an effect on the other

-data are random and independent

-residuals are normally distributed

-homogeneity of variance

-check assumptions at the end of test

Pearson correlation coefficients

-a correlation exists between 2 variables when the values of one variable are somehow associated with the values of the other variable

-a linear correlation exists between 2 variables when there is a correlation and the plotted points of paired data result in a patter than be approximated by a straight line

-r is always between -1 to 1

-r measures the strength of a linear relationship

-r is sensitive to outliers

generic Pearson correlation test null and alternative hypotheses

-H0: rho = 0

-HA: rho =/ (does not equal) 0

Pearson correlation test in R

-cor.test(y ~ x)

--both variables need to be numeric vectors of equal length

parameters of a linear regression model

-y = b0 + bi * x

-b0 = the intercept

-bi = the slope

residuals are the vertical distances of the data to the value predicted by the line

-linear regression fits a line to the data that minimizes the squares of the residuals

interpret R2 and recognize residuals

-R2 = 1 - [sum of (yi -y^)^2 / sum of (yi - ybar)^2]

-y^ = predicted value

-ybar = mean value

-numerator = sum of squared residuals

-denominator = sum of squared distances from mean

-R2 = (variance explained by line) / (total variance)

-R2 states what percent of variance is explained by a certain variable

generic linear regression null and alternative hypotheses

-H0: b1 = 0 (slope of best fit line is 0)

-HA: b1 =/ (does not equal) 0 (slope of best fit line is not 0)

linear regression in R

-lm(y ~ x)

-y = dependent variable

-x = independent variable

-save as model

-summary(model) to see p-value

checking linear regression test assumptions

-check after testing

-plot(model)

-Residual graph: want to see an even cloud across 0

-QQ graph: follow the QQline

what kind of variables do chi-squared test use

frequency counts of categories

what kind of variables do linear regression and correlation use

numerical variables to each other

what kind of data do paired t-test, independent t-test, and ANOVA us

population means to each other

what data does z-test and one-sample t-test use

unknown population mean compared to known population mean or set value