DATA1001

contemporaneous control group

occurs at the same time as the treatment group

Simpson's Paradox

when relationships between percentages in subgroups are different from when the subgroups are combined, because of a confounding variable

placebo

a pretend treatment

placebo effect

the phenomenon in which the expectations of the participants in a study can influence their behavior

Why do we do a controlled experiment?

to focus on the effects of the treatment

reproducible research

a report that can be checked and verified by a third party

double blind experiment

an experiment where both the subjects and investigators do not know which subjects received which treatments

What is the "gold standard" for performing a data study?

randomized controlled trial (RCT)

observational study

a study based on data in which no manipulation of factors has been employed

confounding factors

hidden factors that influence the results

head(data, n) in R

returns the first n rows of data

tail(data, n) in R

returns the last n rows of data

str(data) in R

returns the structure of data

qualitative variable

descriptive variable

quantitative variable

numerical variable

discrete quantitative variable

can take on only particular values and no other values in between

continuous quantitative variable

can take on any value in a given interval of real numbers

ordinal qualitative variable

can be ordered or ranked (e.g. months, days of the week)

nominal qualitative variable

cannot be ordered or ranked (e.g. favorite color)

binary qualitative variable

only two outcomes

graph for 1 qualitative variable

simple bar graph

graph for 2 qualitative variables

double bar graph

graph for 1 quantitative variable

histogram or boxplot

graph for 2 quantitative variables

scatterplot

graph for 1 quantitative variable with multiple categories

comparative boxplot

height of each block in a histogram

(% in block) / (length of interval)

IQR

Q3-Q1

threshold for outliers

Q3 + (1.5 * IQR)

Q1 - (1.5 * IQR)

mean

average - sum of data / size of data

median

middle data point of an ordered dataset

Is the mean robust?

No - easily affected by outliers. Good for symmetric data with not a lot of outliers.

Is the median robust?

Yes - not easily affected by outliers. Good for skewed data.

What should the mean and median be paired with?

they should be paired with a measure of spread

standard deviation

average distance of all the observations of a variable from the mean

standard deviation formula (population)

standard deviation formula (sample)

normal curve

- 68% within 1 std. dev

- 95% within 2 std. dev

- 99.7% within 3 std. dev

standard units (z-score)

how many standard deviations a data point is away from the mean

z-score formula

coefficient of variation

dim(data) in R

returns the dimensions (rows and columns) of data

if we have data size n with sample SD, how do you get the population SD?

SDpop = SDsam * sqrt(n-1 / n)

sd(data) in R

returns the sample SD of data

popsd(data) in R

returns the population SD of data (requires rafalib)

steps to calculate RMS

square the numbers, then mean the result, then root the result (reverse)

How does population SD relate to RMS?

SDpop = RMS of (gaps from the mean)

standard normal curve

normal distribution with mean 0 and SD 1

general normal curve

can have any mean and SD

What goes into an individual measurement?

individual measurement = exact value + chance error + bias

chance error

small differences in sampling due to chance

How to estimate chance error

Replicate measurements and calculate SD

bias (systematic) error

a constant error added to/subtracted from each measurement; can be deliberate or accidental

For a Normal curve with mean 10 and standard deviation 4, what percentage of the data lie between 10 and 14?

34%

qnorm(n, x, SD) in R

gets the nth percentile of a standard distribution with mean x and SD (default values for x and SD are 0 and 1)

pnorm(n, x, SD)

gets the area of the normal curve (with mean x and SD) up to n

notation for general standard curve

X ~ N(x, v) where x is the mean and v is the variance (SD squared)

Assuming we have zero bias, when we take a set of measurements in the real world, we can think of each individual measurement as equal to

exact measurement + chance error

What code would work out the area of a General Normal curve, with mean = 4 and SD = 5, from 0 down.

pnorm(0,4,5)

bivariative data

data with two variables

explanatory variable

a variable that we think explains or causes changes in the response variable, i.e. independent variable (x)

response variable

a variable that measures an outcome or result of a study, i.e. dependent variable (y)

linear association

how tightly the points cluster around a line

strong association

points are clustered close together.

weak association

points are clustered far apart.

positive association

as x increases, y increases

negative association

as x increases, y decreases

correlation coefficient (r)

A numerical index of the degree of relationship between two variables; between -1 and 1.

correlation coefficient formula (population)

for sample r, denominator is n - 1

slope of the SD line

SDy/SDx

slope of the regression line

r(SDy/SDx)

steps to predict y from percentile of x

1. Find z-score of x: Zx = qnorm(percentile)

2. Translate z-score to y: Zy = r * Zx

3. Translate Zy to percentile: pnorm(Zy)

residual

the vertical distance of a point above/below a regression line; represents error between actual value and prediction

RMS error

average gap between the points and regression line

RMS error formula

If a linear model is appropriate, what should the residual plot show?

no pattern

homoscedastic data

the vertical spread of the data is similar over the values of x

heteroscedastic data

the vertical spread of the data is unequal over the values of x

normal approximation within a vertical strip

mean* = mean of y + (Zx x r x SDy)

SD*y = RMS error

prosecutor's fallacy

where it is assumed that the probability of a random match = the probability that the defendant is innocent

conditional probability

the probability that one event happens given that another event is already known to have happened: P(A | B) (probability of event A given event B)

independent events

The outcome of one event does not affect the outcome of the second event

dependent events

The outcome of one event does affect the outcome of the second event

probability of A AND B

P(A) x P(B | A)

mutually exclusive events

events that cannot happen at the same time

probability of A or B if mutually exclusive

P(A) + P(B)

probability of A or B if not mutually exclusive

P(A) + P(B) - P(A and B)

How many different ways to arrange a set of n objects?

n!

If you have a set of n objects and randomly select x objects from the set, how many possible combinations are there?

n choose x

binomial model

Predicting the number of successes in a fixed number of binary trials

If you have n binary trials, with one event having p chance to occur, what is the probability that the event occurs exactly x times?

binomial theorem formula

law of large numbers

as the length of a simulation increases (i.e. sample size increases), the proportion of a certain event approaches a fixed relative frequency

As sample size increases, does the relative error increase?

No - the absolute error will increase, but the error relative to sample size will decrease

box model

a model for describing chance processes

expected value for sample sum

EV = n * mean (n is the sample size)

standard error for sample sum

SE = SDbox * sqrt(n)

expected value for sample mean

EV = mean of box

standard error for sample mean

SE = SDbox / sqrt(n)

SD of a binary box

SE = (x - y)sqrt(% of x * % of y)

central limit theorem

The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution (usually n > 30)

continuity correction

Adjustment made when a discrete random variable is being approximated by a continuous random variable (0.5)