Final (stats) Flashcards
Table
used to present many numerical values
Figure
used to show patterns, trends, or relationships
Qualities of a good table
Should be understandable on its own
Includes appropriate title in proper location
Logical format
Justified numbers → decimal points line up
Good / consistent spacing
Legend
Qualities of a good figure
Understandable on its own
Axes labels (with units)
Appropriate scaling of axis
Symbols
Customized (not the excel default)
No need for box borders around graph
Trendline should be thicker and clear
Figure legends (caption)
The key to understanding a figure
A good figure legend includes:
Title
Materials and methods (description of techniques used)
Results (further explanation of the data)
Definitions (of symbols, patterns, lines, abbreviations, etc).
Monty Hall Problem
It involves a scenario where you have a 1/3 chance of initially choosing the door with a prize behind it. When the host reveals one of the other doors with no prize, the probabilities shift. By switching doors, you essentially capitalize on the new information and increase your chances of winning to 2/3.
Probability
The degree of certainty or chance that something will happen.
Statistics
Help us…
Reduce and describe data
Quantify relationships among data
Determine if sets of data are similar / different
Goals of a data analysis
Data reduction (and description)
Reduce measures to make more meaningful
Averages, spread, bar chart / plots / histograms (descriptive)
Easier and more meaningful to read than all the individual data.
Establish relationships
Descriptive – describe relationship between two observations
Relationship between height and weight
Casual – did something cause the other
Intervention → caused some response
Inference
Infer outcome from sample to population
Is what we see in sample true in population
Purpose of sampling
to approximate a larger population on characteristics relevant to the research question.
Histograms
Graphical representations
Mainly represent frequency (# of subjects that fall into a range).
Measures of central tendency
Mean
average
x̄ = ΣX / N
Median
middle of distribution
Mode
most frequently occurring value
Range
difference between high and low values in a data set
Confidence interval
interval estimate of the population mean (using SEM)
Standard Error of the Mean (equation)
standard deviation / √sample size
Normal distribution
probability that is symmetric about the mean
Kurtosis
measure of outliers in a distribution
High kurtosis → heavy tails or outliers (platykurtic
Low kurtosis → light tails or no outliers (leptokurtic)
Standard deviation
Measure of data around the mean
Amount by which every value varies from the mean
How tightly values in dataset are bunched around the mean
Variability of individual observations around a single sample mean
Central limit theory
when many samples are drawn from a population, the means of these samples tend to be normally distributed.
Empirical rule
for a normal distribution, nearly all data fall within three standard distributions of the mean.
Standard error of the mean (SEM)
how close sample values are to the average of all data points
also shows how accurately the average reflects the sample data
essentially compares the experimental mean to the true sample mean
SEM will always be lower than SDEV
the larger the sample, the lower the SEM which is good
Confidence intervals
give an estimate of how well the sample mean represents the population mean.
Range of likely values for population parameter
Uses reliability coefficient (s) and SEM
Statistical hypothesis testing
Applies the scientific method to data with random fluctuation
The null hypothesis (H0)
effect of data does not represent real effect in hypothesis but is merely a result of random fluctuation.
Hypothesis that there will be no difference nor relationship between variables.
Alternative hypothesis (Ha)
hypothesis formulated based on existing knowledge, theories, or observations.
Difference between variables is specified (one group is greater / uses the other)
One Tailed vs. Two Tailed Test
One-tailed → clear directional prediction based on prior knowledge
Two-tailed → no specific direction expected / both directions equally plausible
The decision to use one tailed vs. two tailed test must be made prior to conducting analysis.
P-Value
The p-value is the probability of finding your result when the null hypothesis is true.
Computed under the assumption the null is supported
Region of high probability = high p-value (shaded blue)
Region of low probability = low p-value (shaded gray)
if the p-value is 0.05 % that tells us that we have a 5% chance of the data supporting the null hypothesis. the confidence level tells us that given this value, we think that there is a 95 % chance that the data will fall into the alternative hypothesis
Parameters of likelihood for observations
called alpha levels
pre determined (usually 0.05)
Alpha value is a probability value
The probability threshold decided is low enough for us to decide the null hypothesis is unlikely to be true.
If p-value is less than alpha, it is unlikely the null is true.
Significant difference
If p-value is low enough, we reject the null hypothesis and conclude a significant difference. (when p is < a → less than a)
Parametric tests
Assumes the sample represents the population
Follows a normal population distribution (regular bell shape)
Non-parametric tests
No assumptions
The area of study is better represented by the median (not the normal distribution)
Very small sample size
Ordinal or ranked data, or outliers cannot be removed.
T-test (overview)
compares the means between two groups
Based on t distribution
T-value measures the size of the difference relative to variation in sample data.
Independent (unpaired) t-test
Grouping categories are independent and unrelated
Ex. different people, animals, or things where values of one group do not affect the other.
Dependent (paired) t-test
Grouping categories are related
Ex. the same person at two points in time.
If the t test statistic is greater than the critical value, the null can be rejected.
A smaller sample means fatter tails (greater likelihood that values will be outliers → bad thing). A larger sample indicates that the value will be closer to the mean.
ANOVA
compares the means among three or more groups
Degrees of freedom
Is the freedom to vary
n (sample) - 1 = degrees of freedom
Indicates the number of independent pieces of information
Critical value
AKA THE Z-VALUE
a specific value or threshold used to determine the acceptance or rejection of a statistical test or hypothesis
Types of error
Type I error (false positive) → rejection of a null hypothesis that is actually true in the population (theres a significance when there actually isn’t)
Type II error (false negative) → failure to reject a null hypothesis that is actually false in the population (theres no significance when there actually is)
Type II is especially a risk when doing multiple t-tests instead of doing ANOVA
Bonferroni technique
You reduce the chances of incorrectly rejecting the null hypothesis (type I error) in any of the individual tests, but it also increases the possibility of making a Type II error (false negatives), meaning that you might fail to detect a true effect.
Looks at the alpha level and divides that by the number of comparisons being made.
Downsides of just an ANOVA test
Doesn’t tell us which means are different, it only determines that there is a difference.
Hence, we can follow that with a post-hoc test.
They help identify where the significant differences lie, providing more specific and detailed information about the relationships between the groups or conditions being compared
Correlation
Measures the association of two variables
Uses correlation when we want to quantify the strength and direction of a relationship.
Depictions of correlations
r < 0.30 → weak to no correlation
r = 0.60 → moderate to strong relationship
r > 0.70 → substantial to very strong relationship
r
correlation coefficient
Refers to a measure of the strength and direction of the linear relationship between two variables. It ranges between -1 and 1, where -1 indicates a perfect negative linear relationship, 1 indicates a perfect positive linear relationship, and 0 indicates no linear relationship. Essentially it tells you strength (magnitude) and direction (positive or negative sign) for the relationship.
The Pearson correlation coefficient is the specific calculated value.
r^2
coefficient of determination
The amount of variance in one variable that is explained or predicted by variance in another variable.
It ranges between 0 and 1, where 0 indicates that the independent variables explain none of the variability in the dependent variable, and 1 indicates that the independent variables explain all of the variability.
b
slope
a
slope
Best straight line fit
Minimize the sum of the squared difference between data and curve fit line.
y = mx + b
Single vs. multiple correlations
Single correlation refers to the relationship between two variables, typically measured using the Pearson correlation coefficient (r-value)
Multiple correlation refers to the relationship between a dependent variable and multiple independent variables.
Type I Error
The probability of committing a type I (rejected the null but it’s actually true) error is whatever our alpha value is set to.
Ex. for an alpha value of 0.05, we have a 5% chance of committing type I error.
Type II Error
The probability of committing a type II error (failure to reject the null when it is false) is denoted by beta, which has to do with power (its typical value is 0.8 → 80% chance we are sure we aren’t committing a type II error).
Beta is the probability that the experiment will yield a not significant result
High power = high chance that your experiment will find a statistically significant result when one is present.
Power
the probability of rejecting the null hypothesis when it is false (good thing).
Is the probability (can’t be negative)…
Of making a correct decision
That a significance test will pick up an effect that is present
Of avoiding a type II error
Effect of Power On:
- effect size
- sample size
- sample variance
- alpha level
Effect size
Large discrimination (shown in bottom right) indicates that we won’t need much power to be sure that the groups are different since they are more spread out and distinct.
Hence, more power is needed to detect smaller differences.
Sample size
Smaller sample sizes require larger amounts of power to detect differences
Sample variance
Higher variance yields small amounts of power.
Alpha value
Alpha level is proportional to power
lowering the alpha value increases the strictness of the test, making it harder to reject the null hypothesis
Diagnostic tests
used to determine the presence or absence of a particular condition in an individual.
Validity is evaluated by a test’s ability to assess the presence (sensitivity) or absence (specificity) of a medical condition.
Tries to answer a yes or no, often from a non-binary variable. Thus, there must be a cut off point to help create a yes or no answer.
True positive
test predicts condition and they have condition
False positive
test predicts condition but they do not have condition
False negative
test predicts no condition but the have condition
True negative
test predicts no condition and they do not have condition
Prevalence
portion of population that has a condition
(true positive + false negative) / everyone (total population)
Sensitivity
proportion of people with conditions that test positive (relative to all individuals with the condition).
True positives / total people with condition (TP + FN)
Specificity
proportion of people without condition that test negative (relative to all individuals without the condition)
True negatives / total people without condition (TN + FP)
Positive predictive value
proportion of people with condition that tested positive for the condition (relative to all p tests).
True positives / total positives (TP + FP)
Negative predictive value
proportion of people without condition who test negative for the condition (relative to all negative tests)
True negatives / total negative tests (TN + FN)
Accuracy
ability to identify true results
(True positives + true negatives) / total number of tests (TP + FP + TN + FN)
Prevalence
proportion of the population that has a condition
(True positive + false negative) / total population
z-value
related to confidence intervals
tells you how many standard deviations you are away from the mean. If a z-score is equal to 0, it is on the mean
how to calculate confidence interval
CI % = x +/- z*(s/√n)
x = sample mean (overall and stays consistent)
z = z-value typically taken from a graph. corresponds to the desired confidence interval %
s = standard deviation
n = sample
output is a range
z-value at each confidence interval
99% –> 2.576
95% –> 1.96
90% –> 1.645
percentage of common association equation
r value * 100
how do t-stat and critical value effect the acceptance or rejection of the critical value
t test statistic is greater than the critical value, the null can be rejected.
