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Biostatistics > Exam 3 > Flashcards

Exam 3 Flashcards

1
Q

What is important to remember about the sampling distribution of means?

A

A population with a normal distribution has a distribution of sample means that is normal

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2
Q

What statistic can you use to test a distribution of sample means? (first one)

A
  • Z standardization, however you rarely know the population standard deviation so substitute and use a students t-distribution
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3
Q

Describe the student’s t distribution.

A
  • similar to standard normal distribution (z) but with fatter tails
  • as the sample size increases, the t distribution becomes more like the standard normal distribution
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4
Q

What can the t-distribution be used for?

A
  • It can be used to accurately calculate a confidence interval for the mean of a population with a normal distribution
  • (population mean) - (Tcritical value x SE(standard error of mean) )< (actual mean) < (population mean) + (Tcritical value x SE(standard error of mean) )
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5
Q

What is the standard error of the mean?

A

SE (y) = s / sqrt(n)

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6
Q

What is a one sample t-test?

A

compares the mean of a random sample from a normal population with the population mean proposed in a null hypothesis

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7
Q

What are the hypotheses and test statistic in a one sample t test?

A

H0 - true mean equals u0
Ha - true mean does not equal u0

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8
Q

How do you interpret the t-statistic in a one sample t-test?

A
  • compute the p value: probability of this t-statistic or more extreme given the null hypothesis is true
  • if p value is >.05 then you fail to reject the null hypothesis
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9
Q

How does increasing sample size affect a one sample t test?

A
  • increasing sample size reduces the standard error of the mean
  • increase the probability of rejecting a false null hypothesis (power)
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10
Q

What are the assumptions of a one-sample t-test?

A
  • data are a random sample from the population
  • variable is normally distributed in the population (robust to departures)
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11
Q

What are the confidence intervals for variance and standard deviation? And what are the assumptions of these statistics?

A

Assumptions: random sample from the population, variable must have a normal distribution (formulas are NOT robust to departures from normality)

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12
Q

What are two different study designs when comparing two means?

A

two-sample and paired designs

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13
Q

What is a two sample design?

A
  • two groups
  • each group is composed of independent sample of units
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14
Q

What is a paired designs?

A
  • two groups
  • each sampled unit receives both treatments
  • paired designs are usually more powerful because of control for variation among sampling units
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15
Q

How are paired designs treated?

A
  • paired measurements are converted to a single measurement by taking the difference between them
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16
Q

What is a paired t-test?

A
  • used to test the null hypothesis that the mean difference of paired measurements equals a specific value
  • null is often that the difference (change) is zero before and after treatment
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17
Q

How does a paired t-test compare to a one sample t-test?

A
  • The same process except the calculation of the test statistic occurs on the difference value (d)
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18
Q

What does the p-value indicate in a paired t-test statistic?

A
  • P-value >0.05
  • Fail to reject the null hypothesis that the mean change is zero
  • P-value <0.05
  • reject the null hypothesis that the mean change is zero
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19
Q

What are the assumptions of a paired t-test?

A
  • sampling units are randomly sampled from the population
  • paired differences have a normal distribution in the population
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20
Q

What test is a formal test of normality?

A

the shapiro-wilk test

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21
Q

What are the hypotheses of the shapiro-wilk test? Why should it be used with caution?

A
  • H0 = sample has normal distribution
  • Ha = sample does not have normal distribution
  • Should be used with caution:
  • small sample sizes lack power to reject a false null (Type 2 error)
  • large sample sizes can reject null when the departure from normality is minimal and would not affect methods that assume normality
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22
Q

Under the null hypothesis, the sampling distribution of the one-sample t statistic follows a _________

A

t distribution with n-1 DOF

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23
Q

Describe the t distribution.

A

-The area under the curve to the left (lower tail) of -t is the same as the area to the right (upper tail) of t
- t distribution is symmetrical around the mode of zero

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24
Q

What does a Shapiro Wilk test evaluate?

A
  • evaluates the goodness of fit of a normal distribution to a set of data randomly sampled from a population
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25
Q

What is something other than a t-test that you may be asked to do with paired measurements?

A
  • Can calculate the 95% CI for true mean difference
  • If the range includes zero (no difference)
  • other options: may be consistent with a decrease or increase
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26
Q

How do you find the confidence interval for difference between two means (two sample t-test)?

A
  • statistic of interest (mean 1 - mean 2)
  • calculate pooled sample variance
  • then calculate confidence interval for difference between two means
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27
Q

What is pooled sample variance and what is it used for?

A
  • the averaged of the variances of the samples weighted by their degrees of freedom
  • used for calculating confidence interval for the difference between means in two sample test
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28
Q

How do you compare the means in a two sample test?

A

a two sample t-test

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29
Q

What does a two sample t-test do? What are the null and alternative hypotheses?

A
  • compares the means of a numerical variable with two independent groups
  • H0: mean 1 = mean 2
  • Ha: mean 1 does not equal mean 2
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30
Q

What are the assumptions of a two sample t-test?

A
  • each of the two samples is a random sample from its population
  • numerical variable is normally distributed in each population
    (robust to minor deviations)
    -standard deviation and variance of the numerical variable is the same in both populations (robust to some deviations if the sample sizes are approximately equal)
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31
Q

What is the formal test of equal variance? What are the hypotheses?

A
  • levenes test
    H0: variances are equal
    Ha: variances of the two groups are not equal
    Can be extended to more than two groups
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32
Q

What do you do to compare the means in a two sample t-test if the variances in the two groups are not equal?

A
  • standard t-test works well if both sample sizes are greater than 30 and there is a less than 3 fold difference in standard deviations
  • Welch’s t-test can be used even when the variances of the two groups are not equal - slightly less power compared to the standard t-test
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33
Q

What is Welch’s t-test?

A
  • Welch’s t-test compares the means of two groups and can be used even when the variances of the two groups are not equal
  • slightly less power compared to the standard t-test
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34
Q

What is important with units when comparing means of two groups?

A
  • correct sampling units
  • when comparing means of two groups an assumption is that the samples being analyzed are random samples, but often repeated measurements are taken on each sampling unit
  • fish in streams example, proportion of fish surviving in each stream
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35
Q

What is the fallacy of indirect comparison?

A
  • compare each group mean to hypothesized value rather than comparing group means to each other
  • since group 1 is significantly different than zero, but group 2 is not then groups 1 and 2 are significantly different from each other
  • comparisons between two groups should be make directly, not indirectly by comparing group to the same hypothesized value
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36
Q

What are the four potential options to address violations of assumptions?

A
  • Ignore the violations
  • Transform the data
  • Use a non-parametric method
  • Use a permutation test
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37
Q

What is a plot used to detect deviations from normality?

A

normal quantile plot
- may also used frequency distribution histograms, box plots, and strip plots - but sample distribution may look noisy due to sampling error

38
Q

What is a normal quantile plot?

A

compares each observation in the sample with its quantile expected from the standard normal distribution. Points fall roughly along a straight line if the data come from a normal distribution

39
Q

When can you ignore violations of normality for t-tests? When should you not?

A
  • When sample sizes are large the sampling distribution of means behaves roughly as assumed by t-distribution
  • Large sample size depends on the shape of the distribution:
    —If distribution of two groups being compared are skewed in different directions, then avoid t-test even for large samples
    —If distributions are similarly skewed then there is more leeway
40
Q

When can you ignore violations of equal variance/sd for two sample t-tests?

A

Two-sample t-tests assume equal standard deviations in the two populations
- If samples sizes are > 30 in each group and sample sizes in two groups are even, then even up to a 3x difference in standard deviation can be ok
- Otherwise use Welch’s t-test

41
Q

What is important to know about data transformations?

A
  • A data transformation changes each measurement by the same mathematical formula
  • Can make standard deviations more similar and improve the fit of the normal distribution to the data
42
Q

What are common uses of log transformations?

A

o Measurements are ratios or products
o Frequency distribution skewed to the right
o Group having larger mean also has larger standard deviation
o Data span several orders of magnitude

43
Q

What are five other data transformations and their uses?

A
  • Arcsine (best use: proportions)
  • Square-root (counts)
  • Square (skewed left)
  • Antilog (skewed left)
  • Reciprocal (skewed right)
44
Q

What are non-parametric alternatives to t-tests?

A
  • A nonparametric method makes fewer assumptions than standard parametric methods do about the distributions of the variables
  • Can be used when deviations from normality should not be ignored, and sample remains non-normal even after transformation
  • Don’t rely on parametric statistics like mean, sd, s^2
  • Usually base don the ranks of the data points rather than the actual values
45
Q

What are three non-parametric tests?

A
  • sign test (paired/one sample)
  • wilcoxian signed-rank test (paired)
  • Mann-Whitney U-Test (two-groups)
46
Q

What is a sign test? When is it used? How does it work? What are the drawbacks? What must be remembered?

A
  • The sign test compares the median of a sample to a constant specified in the null hypothesis. It makes no assumptions about the distribution of the measurement of the population
  • Each measurement is characterized as above (+) or below (-) the null hypothesis
  • If the null is true, then you expect half the measurements to be + and half to be –
  • Uses binomial distribution to test if the proportion of measurements above the null hypothesis is p=0.5
  • Can be replacement of one-sample or paired t-test
  • Low power compared to t-tests: Impossible to reject null when the sample size is <5
  • Typically you remove data points that indicate that the value is exactly null (difference of zero), this reduces n
47
Q

What is a Wilcoxian signed rank test?

A
  • Non-parametric sign test for paired/one sample t-test
  • More power than standard sign test because information about the magnitude away from the null for each data point
  • But test assumes that population is symmetric around the median (i.e., no skew)
  • Nearly as restrictive as normality assumption, thus NOT recommended
48
Q

What is the nonparametric test for two samples?

A

Mann-Whitney U-Test

49
Q

What is the Mann-Whitney U-test?

A
  • Nonparametric test for two samples
  • The Mann-Whitney U-Test compares the distribution of two groups. It does not require as many assumptions as the two sample t test
    H0: the means are the same
    HA: the means are different
    Functions through ranking of sums
  • Use Mann-Whitney U-distribution to calculate p-value
50
Q

What happens in Mann-Whitney U-test if you have tied ranks?

A

Assign all instances of the same measurement the average of the ranks that the tied points would have received

51
Q

What are the assumptions of nonparametric tests? Assumption for all? Wilcoxian? Mann-Whitney U?

A
  • Still assume that both samples are random samples from their populations
  • Wilcoxian signed-rank test assumes distributions are symmetrical (big limitation – not recommended)
  • Rejecting null hypothesis of Mann-Whitney U-test means two groups have different distributions of ranks, but does not necessary imply that means or medians of groups differ
  • To make this inference there is an assumption that the shapes of the distributions are similar
52
Q

What are the assumptions of a paired t-test?

A
  • Sampling units are randomly sampled from the population
  • Paired differences have a normal distribution in the population
53
Q

What is the problem with comparing multiple means at once?

A
  • It is tempting to do all possible pairwise comparisons
  • But the problem is that running multiple tests inflates the probability of getting at least one Type I error
54
Q

What does ANOVA do?

A
  • Analysis of Variance (ANOVA) compares the means of multiple groups simultaneously in a single analysis
  • Tests for variation of means among groups
  • Anova determines if there is more variance among sample means than we would expect by sampling error alone
55
Q

What are the hypotheses of ANOVA?

A

o H0: mean 1 = mean 2 = mean 3
o Ha: mean of at least one group is different from at least one other group
- Null assumption that all groups have the same true mean is equivalent to saying that each group sample is drawn from the same population

56
Q

What are anova’s two measures of variation?

A

Group mean square and error mean square

57
Q

What is group mean square? MSgroups

A
  • proportional to the observed amount of variance among group sample mean
  • variation among groups
58
Q

What is error mean square? MSerror

A
  • estimates the variance among subject that belong to each group
  • variation within groups
59
Q

What is the specific null and false null for anova?

A
60
Q

What are the two sums of squares in anova?

A
  • SSgroups= calculates sources of variation among groups
  • SSerror= calculates sources of variation within groups
61
Q

What is the F-ratio test statistic in anova? DOF?

A
  • F-statistic increases as the differences among sample means for treatment groups increases
  • The F statistic has a pair of degrees of freedom
  • Numerator: k -1
  • Denominator: N – k
62
Q

What is R^2 for anova?

A

measures the fraction of variation in Y that is explained by group differences

63
Q

What are the assumptions of Anova tests?

A
  • Measurements in every group represent a random sample from the corresponding population
  • Variable is normally distributed in each of the populations
    o Robust to deviations, particularly when sample size is large
  • Variance is the same in all k populations
    o Robust to departures if sample sizes are large and balanced, and no more than 10x differences among groups
64
Q

What is a nonparametric alternative to an ANOVA?

A

Kruskal-Wallis test

65
Q

What is a planned comparison?

A
  • A planned comparison is a comparison between means planned during the design of the study, identified before the data are examined
  • In circadian clock follow-up study, the planned (a priori) comparison was difference in means between knee and control group
  • Use SE and CI calculations for planned comparisons
66
Q

What is an unplanned comparison? What test is used?

A
  • Comparisons are unplanned if you test for differences among all means
  • Problem of multiple tests (increasing probability of Type I error) should be accounted for
  • With the Tukey-Kramer method the probability of making at least one Type I error throughout the course of testing all pairs of means is no greater than the significance level alpha
  • An unplanned comparison analysis applies to all mean comparisons among groups. The analysis is typically done with the Tukey-Kramer method, in which it is harder to reject the null hypothesis compared to a two-sample t-test.
67
Q

How does the Tukey-Kramer method function?

A
  • Works like a series of two-sample t-tests, but with a higher critical value to limit the Type 1 error rate
  • Because multiple tests are done, the adjustment makes it harder to reject the null
68
Q

What is the Kruskal-Wallis post-hoc test?

A

-Suppose your data:
o Fail normality even after transformation
o Generate a significant Kruskal-Wallis result
-So the interpretation is that the distribution of ranks differs for at least one group. But which one?
-Should not use Tukey-Kramer, which is a parametric test
- Dunn’s test is the appropriate analysis for a post-hoc analysis of groups following a significant Kruskal-Wallis result: Will compare all possible pairs of groups while controlling for multiple tests

69
Q

Which sums of squares ?

A

total

70
Q

Which sums of squares?

A

error

71
Q

Which sums of squares?

A

group

72
Q

What is correlation?

A

When two numerical variables are associated then they are correlated

73
Q

What is the correlation coefficient? What does it measure? Range? What does it depend on?

A
  • The correlation coefficient measures the strength and direction of the association between two numerical variables
  • Ranges from -1 to 1
  • Possible that two variables can be strongly associated but have no correlation (r=0) – example is a non-linear association
  • Correlation coefficient depends on range
74
Q

What are the two different correlation coefficients?

A

Correlation coefficient (statistic) r
Population correlation coefficient (parameter) p

75
Q

What is important about the standard error of the correlation coefficient?

A
  • Standard error of correlation coefficient
  • Can be calculated, but the sampling distribution of r is not normally distributed, so standard error of r is not used in calculating the 95% CI
76
Q

What is the process of hypothesis testing for correlation?

A
  • H0: p = 0
  • HA: p does not equal 0
  • Use standard error and get a test statistic, then use students t distribution
77
Q

What are the assumptions of hypothesis testing for correlation?

A
  • random sample from the population
  • bivariate normal distribution:
    bell shaped in two dimensions rather than one
78
Q

What are three deviations from bivariate normality?

A
  • funnel
  • outlier
  • non linear
79
Q

What is regression?

A
  • Regression is a method that predict values of one numerical variable from values of another numerical variable
  • Fits a line through the data
    o Used for prediction
    o Measures how steeply one variable changes with the other
80
Q

How does linear regression function?

A

Draws a straight line through the data to predict the response variable (y, vertical axis) from the explanatory variable (x, horizontal axis)

81
Q

What is the least-squared regression line?

A
  • line for which the sum of all the squared deviations in Y is smallest
  • Y = a +bx
  • A is the Y intercept; b is the slope
  • The slope of a linear regression is the rate of change in y per unit x
  • Once slope is calculated, getting intercept is straightforward because the least-squares regression always goes through the point (mean X, mean Y)
82
Q

Samples vs populations least squared regression line. What does regression assume?

A
  • The slope (b) and intercept (a) are estimated from a sample of measurements, hence these are estimates/statistics
  • The true population slope (beta) and intercept (alpha) are parameters
  • Regression assumes that there is a population for every value of X, and the mean Y for each of these populations lies on the regression line
83
Q

When predicting with regression line what are you predicting?

A
  • Predictions are mean Y for all individuals with value X
  • Designated Y-hat
84
Q

What are residuals? What measures their scatter?

A
  • The residual of a point is the difference between its measured Y value and value of Y predicted by the regression line
  • Can be positive or negative
  • Variance in residuals (MSresidual) quantifies the spread of the scatter
    —Residual mean square
    —Analogous to error mean square in Anova
    —Used to quantify the uncertainty of the slope
85
Q

How do you calculate the error in slope of a regression line?

A
  • Standard error of slope – uncertainty (precision) with the sample estimate b of the population slope beta
  • Can be used to calculate confidence interval of slope
86
Q

What are two types of predictions you can make with a linear regression line? Which is more precise?

A

Predict mean Y for a given X
o Confidence bands measure the precision of the predicted mean Y for each value of X
o What is the mean age of all male lions whose noses are 60% black?
Predict single Y for a given X
o Prediction intervals measure the precision of the predicted single Y-values for each X
o How old is that lion over there with a 60% black nose
Both predictions give the same value of Y, but they differ in precision
o Can predict mean with more certainty that a single value

87
Q

What should regressions be used for?

A

interpolation, not extrapolation

88
Q

How do you perform hypothesis testing on a linear regression slope?

A
  • H0: age cannot be predicted by proportion of black on nose (beta = 0)
  • HA: age can be predicted by proportion of black on nose (beta does not equal 0)
89
Q

What is R^2 for linear regression?

A

the fraction of variation in Y that is explained by x

90
Q

What is regression towards the mean?

A

results when two variables measured on a sample of individuals have a correlation
less than one. Individuals that are far from the mean for one of the measurements will, on average, lie closer to the mean for the other measurement

91
Q

What are the assumptions of linear regression?

A

At each value of X:
* there is a population of Y-values whose mean lies on the regression line
* the distribution of possible Y-values is normal (with same variance)
* The variance of Y-values is the same at all values of X
* the Y-measurements represent a random sample from the possible Y- values

Biostatistics (4 decks)

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