Inferential statistics

Question 1

Outlier

Answer 1

An outlying observation, or outlier, is one that appears to deviate
markedly from other members of the sample in which it occurs.
different measures:
• more than 3 SD away from the mean
• more than 1.5 times of the IQR (mild) 3 times
(extreme) ñ custom boxplot criteria

Question 2

Confidence Interval (CI)

Answer 2

Confidence interval
we need to quantify uncertainity about the population value
• a confidence interval states our uncertainty
• confidence intervals are available for means,
differences between means, proportions, correlations…
A condence interval is a range of values surrounding the sample estimate that is likely to contain the population parameter. Vorhersage liegt bei 0.25. Wenn CI=0.95 würde würde die Vorhersage in 95 von 100 Fällen den realen Wert abdecken
CI95% = mean+/- 2SE

Question 3

Ho

Answer 3

• H0 with one group:
- our sample proportion is not different from the
known population proportion
- our sample mean is not different from the known population mean
• H0 with two or more groups:
- groups are not different from each other regarding their population proportions or means
• alternative hypothesis: same sentences but without not

Question 4

p-value

Answer 4

gives the probability that we observe a difference as large or even larger as seen with the sample if H0 would be true!

• Can we reject the null hypothesis?
• We will never know if the null hypothesis is true or false!!
• We almost always observe a difference!!
• One group: We need to know the population
mean/proportion!: Tells us how sure we are that our sample is not(!) different from the population
• Two groups: Tells you how often you would get the observed or a larger difference by random sampling from two populations if the means/proportions of both populations would be equal
• The p-value tells us not(!!) how sure we are that there is really a difference

Question 5

CI vs p-value

Answer 5

• p-values are somehow only a measure of randomness
• CI’s tells you about your proportions and the probable value in the population
• CI’s are often the better measure, but unfortunately in science less frequently used
• may be because telling one number is easier than two?

Question 6

Significance

Answer 6

• statistically significant does not necessary mean that
the observation is\important”
• just a custom threshold () for the p-value to claim
significance (mostly < 0.05)
• significant , highly significant **, extremly
significant **

Question 7

Testing Numeric vs Numeric
Numeric vs Categoric
Categoric vs. Categoric

Answer 7

• Numeric vs Numeric
- Correlation, Regression
• Numeric vs Categoric - t-test, anova (today)
• Categoric vs Categorical - chisq-test, sher-test

Question 8

Central Limit Theorem

Answer 8

No matter how the population is distributed: the population of sample means will approximate a Gaussian distribution if the sample size is large enough

What is large ? It depends:
• a less normal distribution more samples (100 should be enough in any case)
• more normal distribution (10 or more)

Question 9

Properties of a Normal Distribution

Answer 9

• symmetrical bell shaped distribution
• extends in both directions to infinity
• mean and median are closed to each other
• 95% of all values are within 2 SD
• this assumption gives very wrong results if the the distribution is non-normal !!
• normal data --> t.test
• non-normal data, skewed or multi-modal
distributions --> wilcox.test

Question 10

t distribution

Answer 10

• Derives from the normal distribution

* t is the difference between the sample mean and the population mean, divided by the SEM

Question 11

Effectsize: Cohens D

Answer 11

How large is the deviation between two groups in comparison to the standard deviation.

Question 12

Correlation

Answer 12

• observe the association between two numerical
variables
• if two numerical variables are associated we say they are correlated
• the correlation coefficient is a quantity that describes the strength of the association

Question 13

4 interpretations of r

Answer 13

Why the variables correlate so well?
• Lipid content of membranes determines insulin sensitivity?
• Insulin sensitivity affects lipid content?
• Insulin sensitivity and lipid content are controlled by
third factor?
• There is no correlation, our r is just a random finding (type 1 error)?
• We did not know the truth …
• Correlation did not mean causation!!!

Question 14

Correlation: r-squared r^2

Answer 14

• r2 often also called coefficient of determination
• r is between -1 and 1
• r2 is between 0 and 1, smaller than r
• r2 is interpreted as the fraction of variance that is
shared between the variables
• runners: 0,192 = 0,036 meaning that only 3.6% or
the variance of time are shared by age
• students: 0,782 = 0,6084 means that 60% of the weight variance is shared by size

Question 15

influence outliers

Answer 15

• Just one point can change everything with Pearson correlation!

Question 16

Spearman Rank Correlation

Answer 16

• Spearman correlation is more robust against outliers!
• Correlation with one outlier is not significant!!
• Spearman correlation is calculated on ranks of values.
• It’s a non-parametric test.
• It does not assumes normal distribution of data.
• It is more conservative.
• If in doubt use Spearman correlation.

Question 17

Effectsize Pearsons r and Spearmans rs

Answer 17

• Pearsons r and Spearmans rs are quite similar in their values
• but rs^2 is the proportion of rank variances for
• Kendalls tau is numerical different
66-75% of r or rs, don’t square it
• r of 0.1 small effect, 1% of variance
• r of 0.3 medium effect, 9% of variance
• r of 0.5 large effect, 25% of variance

Question 18

partial correlation

Answer 18

In probability theory and statistics, partial correlation measures the degree of association between two random variables, with the effect of a set of controlling random variables removed. If we are interested in finding whether or to what extent there is a numerical relationship between two variables of interest, using their correlation coefficient will give misleading results if there is another, confounding, variable that is numerically related to both variables of
interest.
• partial correlation of body height and weight after removing the effect of sex
• correlation of shoesize and writing capabilites after removing the effect of …

Question 19

Mutual Information

Answer 19

• Berechnet die Zahl einer bestimmten Abfolge zur Zahl aller Abfolgen (beispielsweise die Anzahl aller AC-Paare im Vergleich zu allen Paaren)

o Gibt Korrellationsinformationen wie die statistische Unabhängigkeit über die Sequenz für alle Pärchen korrelliert
o Nennt man übrigens auch Boltzmann Entropie, …

Question 20

Correlation vs Regression

Answer 20

Correlation
• description of an undirected relationship between two or more
variables
• how strong it is
• direction is not known, not existing or we are simply not interested
• phones in household and baby deaths

Regression
• description of a directed relationship between two or more variables
• one variable influences the other
• smoking and cancer
• weight and height
• model to describe the relationship
• model to predict one
variable

Question 21

Regression; aims, types;

Answer 21

• looking for a trend: linear, sigmoid, exponential
• curve fitting : which model ist most similar to the data
• prediction: predict response variable Y from X
• standard curve: assays

Regression Types
• simple linear regression (numerical variables)
• multiple linear regression (numerical variables)
• logistic regression (Y is categorical)
• non-linear regression (numerical variables)
• regression trees (Y is numerical)
• classification trees (Y is categorical)

Question 22

Simple linear regression

Answer 22

Simple Linear Regression
• most common regression type
• method to find a best straigth line to a cloud of data points
• one variable (independent) is used to predict a second
(dependent) In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable
and one dependent variable (conventionally, the x and y coordinates in a Cartesian coordinate system) and finds a linear
function (a non-vertical straight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent variables. The adjective simple refers to the fact that the outcome variable is related to a single predictor

Question 23

Regression predict values

Answer 23

Use the equation to determine Y for certain values of X. Example what would be the values of insuline response for C20.22 values of 0, 15,16 and 17%.

Question 24

Linear regression: slope and intercept

Answer 24

Intercept (a, alpha): Value of Y if X is zero (Y-Intercept).
Slope (b, beta )): Increase on Y by one unit of X.
Example: y = 2x + 1
beta= (Summe (xi- durchschnittx)*(yi-durchschnitty))/(summe(xi- durchschnittx)^2)

Question 25

Residual in regression (and error)

Answer 25

Error (or disturbance) of an observed value is the deviation of the observed value from the (unobservable) true value of a quantity of interest (for example, a population mean).
Residual of an observed value is the difference between the observed value and the estimated value of the quantity of interest (for example a sample mean) (Wikipedia).

Question 26

When use a non-parametric test?

Answer 26

• outcome is a rank or score
• some values are “too high” or “too low” - off scale to measure
• measurements although numbers are just not Gaussian
• you can try to transform the data
• log-transformation is often used

Question 27

Tests with gaussian distribution and large vs small size

Answer 27

Non-gaussian
large:parametric ok
but what is large ?
How far from the normal distribution
small: parametric not ok; P-value wrong !

Gaussian

large: non-parametric ok; p-val slightly too large
small: non-parametric not ok, p-value much too high; lack of statistical power

Question 28

(log2/10) Transformation der Daten

Answer 28

Wenn sich Deine Daten als nicht normalverteilt herausstellen, kannst Du versuchen, sie durch Transformation in eine annähernde Normalverteilung umzuformen. Wenn das gelingt, rechnest Du anschließend die weiteren Analysen wie Signifikanztests mit den transformierten Daten. Dann ist es möglich, parametrische Methoden, die Normalverteilung fordern, anzuwenden.

Auch andere Probleme mit der Verteilung, wie zum Beispiel Hetereskedastizität, Nicht-Linearität oder Ausreißer können eventuell mit Transformationen behoben werden.
• natural log2, log10 logarithm are most often used
• values of zero are existing? Add to all values a 1: log(n+1)
• negative values: asinh transformation
• if used un-transform for instance condence intervals for
reporting

Question 29

Multiple testing correction

Bonferroni-Korrektur

Answer 29

• korrigiert die Family-Wise-Error-Range (FWER)
  o FWER = Wahrscheinlichkeit, dass einer oder mehrere Tests falsch positiv sind
• Entweder alle p-Werte mit N multiplizieren (und nur was dann noch immer über 0.05 liegt ist positiv)
• … oder neues alpha berechnen= altesalpha/Anzahl Hypothesen

Question 30

Multiple testing correction Holm-Verfahren

Answer 30

• Kontrolliert noch immer die FWER
• P-Werte der Größe nach sortieren
• Festlegung mehrerer Signifikanzschwellen
  für kleinsten p Wert: alphaNeu= altesAlpha/N
  für zweitkleinsten: alphaneu=altesAlpha/N-1
  …
• Abgelehnt werden Nullhypothesen mit allen p-Werten die kleiner als die zugehörige Schranke sind (bis die Schranke zum ersten Mal überschritten wurde)

Question 31

Multiple testing Benjamin-Hochberg-Verfahren

Answer 31

Kontrolliert die FDR
- p-Werte sortieren
- Festlegung mehrerer Signifikanzschwellen:
für kleinsten pWert: 
alphaneu=alpha/N

für zweitkleinsten:
alphaneu=2xalpha/N

für drittkleinsten:
alphaneu=3xalpha/N

Vorteil gegenüber Holm: Hier können auch nach der ersten Grenzüberschreitung weitere p-Werte abgelehnt werden

Question 32

Visulizins p values

and random p values

Answer 32

histogram

rn=rnorm(100,mean=10,sd=2)