exam questions

1. Correlation analyses can test for linear relationship between variables

2. correlations do not prove causal relationships

3. correlation analyses measure the change of one variable with another

1. there is a strong negative correlation between the two variables

2. 90% of the variance within the data can be explained by the model

3. the correlation is statistically significant

• analyses, in which a third variable is controlled for, while correlating two others. the third variable has an influence on only one of the others variables

• with a high number of observations even correlations with a low-r-value can become significant

• since causalities cannot be identified, if its not possible to identifie, which variable is influencing which

• it can never be ensured that a third variable does not influence at least one of the two observed variables

1. ALWAYS think about your statistics before you collect data.

2. CHECK your data for outliers, normality, correlations, missing values, etc.!

3. Participate in stats-courses whenever possible, refresh your knowledge! PRACTICE!

1)      PLOT your data

2)      DRAW the model

3)      CHECK for:

a)      homogeneity of variances

b)      normality of errors

c)      autocorrelation between residuals

d)      influential data points

4) INTERPRET your model

-          observing patterns

-          deriving models

-          formulating hypotheses

-          designing an experiment/observational study

-          collecting data

-          visualising/analysing the data

model = possible explanation for a driving factor of a pattern

pattern = decrease in chaos; requires energy & driving factor

unexplained variation = variation around the means (= noise)

• Within group variability

explained variation = variation of the means (= signal)

• between group variability (difference between groups)

signal can be covered by the noise (-> too much unexplained variation)

accuracy = measurements are as close to the actual value as possible

precision = repeated measurements are as close to each other as possible

Precision has to do with the resolution and quality of devices we use to obtain data, accuracy with the way we calibrate them.

What you want is high accuracy (strong signal) and high precision (low noise) to obtain a high test power.

test power is the likelihood to find an effect (, if there is one

-          has different levels:

-          interval-data contain the highest amount of information, the highest test power

-          Example: limpets in intertidal horizons

independent variables = what you manipulate, also explanatory variable, predictor variable, factor

dependent variables = what you measure, also response variable, explained variable

-          dependent v. should be a function of independent v.

-          can be categorical or continuous

-          type of variables determines which statistic to use

population = totality of all units characterised by a variable

sample = analysed part of the population

A population corresponds to the real world (unmeasurable), whereas a sample is an approach to describe the real world.

unit = sampling unit = replicate = parallel

statistical population = population of sampling units

treatment level = experimental group

Replicates…

-          need to be independent

-          shouldn’t be repeated measures

-          shouldn’t be grouped together at one place

-          should be of an appropriate spatial scale

Else you get pseudoreplication!

statistic = measure of some attribute of a sample, f.e. the sample mean, estimates a parameter (population mean)

parameter = measure of some attribute of a population, f.e. the population mean

mean =                    , influenced by outliers

median = the observation that has equal numbers of observations above and below it, good for non-normal data

-          are separated into four quartiles (25% of the data points each)

-          show the interquartile range (Q3 - Q1) as a box

-          are the best way to summarise your data graphically, because they indicate distribution of the data

-          good for non-normal data (use the median)

Variance = sum of squared distances from mean / degrees of freedom

Measures how far a set of numbers (replicates) are spread out from their mean (variability around the mean).

Standard deviation:

Measures the amount of variation of a set of data values.

Software uses variances for calculating. We use SD for communicating.

Standard error (of the mean):

Estimates the reliability of a sample statistic (most commonly of the mean). It is the standard deviation of sample means.

Driven by variability of population and number of replicates (n).

How to calculate the standard error:

calculate means of samples -> calculate mean of means -> SD for mean of means = SE

Indicates the precision of an estimated parameter (f.e. population mean).

It gives you the probability (most often 95%, confidence level) with which the true population parameter lies inside the borders of the calculated interval.

Factors affecting the width of the confidence interval are sample size, confidence level and variability in the sample.

-          in a normal distribution 95% of all values are in the range of 1.96 * SD

-          t = correction factor for small samples (n < 100)

-          smaller than 1: study is insignificant

-          the bigger the test statistic, the higher the test power

-          it is influenced by the effect size, the unexplained variation and sample size

• can only handle two samples

1)      avoid noise

2)      increase effect size

3)      increase sample size

It is a continuous probability distribution that is strongly related to the standard normal distribution and was developed to deal with low sample sizes. The difference is that it does not relate to the whole population, but only to a sample. This is because the population standard deviation is almost always unknown, so the sample standard deviation is used instead. Thus, there are multiple t distributions (/samples) to a population and all of them have a higher variance than the standard normal distribution (except for a t-distribution with ).

To construct a 95% confidence interval for a normal distribution, the t-value is 1.96 (). For a t-distribution it will be greater than 1.96, due to the distribution’s greater variance.

1. Type I Error: Rejecting the H0 although it’s right. We believe there is an effect, but there is not. > We see something, where there is nothing.

2. Type II Error: Accepting the H0 though it’s wrong. We believe there is no effect, but there is one. > We overlook the effect.

A directional hypothesis is called one-tailed.

A non-directional hypothesis is two-tailed

Advantage:

> you increase test power

Disadvantage:

> you’ll miss the effect in case

it goes in the other direction

Correlations quantify how strongly two variables covary with each other.

-> r

-          measures the degree of correlation

-          ranges from -1 to +1 | perfect negative to perfect positive correlation

Most common one:

Pearson correlation coefficient:

= measure of the linear correlation between two variables x and y

Spearman’s rank correlation coefficient:

= measure of the correlation between the ranking of two variables

-          requires ordinal data (ranks)

Reveal the unique variance explained

by one variable, while controlling a third

variable.

univariate analysis

●       ANOSIM (analysis of similarity)

●       PERMANOVA (permutational multivariate analysis of variance)

●       ordination techniques and related methods, like PCA or MDS

• express mechanistic understanding of explanatory variable

• should be:

  • accurate

  • convenient

  • adequate (explain sufficient amount of data)

  • minimal (high explanatory power) —> estimating as few parameters as possible

• can contain

  • categorial factors

  • interactions btw factors

  • continuous covariates

minimise error term

They follow the straight line equation:                         

variables

Error term is quantified by the residuals:

The term general linear model (GLM) usually refers to conventional linear regression models for a continuous response variable given continuous and/or categorical predictors. It includes multiple linear regression, as well as ANOVA and ANCOVA (with fixed effects only).

The term generalized linear model (GLM) refers to a class of models in which the response variables have error distributions that do not follow the normal distribution, but an exponential family distribution (f.e. the Binomial-, Poisson- or Gamma-distribution).

• In regression analysis we can test for the influence of one or more continuous variables on one dependent variable.

• in SLR:  is a regression model with one response and one explanatory variable (both continuous).

The best fitting line should be the one that minimizes the sum of the absolute values of the residuals. Though it actually is the line that has the lowest sum of squared residual differences. Thus, the estimate gets less prone to errors.

total sum of squares (SST) = squared differences between the mean and the observed values

SSM = SST – SSR

SSM = SST – SSR

(SSR) = squared differences between the line of best fit and the observed values

(SSM) = the increase in accuracy by replacing the simplest model by the best fit model

SSM = SST – SSR

Coefficient of determination (R2) is the fraction of variance in the dependent variable that’s explained by the independent variable(s)

| R2 * 100 = explained variation in percent

Pearson’s correlation coefficient (r or R) is a measure of the linear correlation between two variables X and Y

The F-ratio is a test statistic that measures the ratio of the variation explained by the model and the difference between the model and the observed data (unexplained variation) by using mean squares (MSM & MSR)

The adjusted R2 takes into consideration how many predictors are in the model and how many data points are in the data set.

For linear regression, we assume that…

-          residuals are normally distributed,

-          the variance in the residuals is constant,

-          the residuals are not correlated,

-          there are no influential data points.

A one-way analysis of variance (ANOVA) compares means of three or more samples to each other. It tests for the influence of a categorical independent variable on a dependent variable.

A series of pairwise t-tests does the same, but it would increase the probability of a type I error and also the number of hypotheses.

ANOVA = compares the variability between samples with the variability within samples

Sample means and underlying population means do not differ from each other.

An ANOVA partitions the total sum of squares into two components:

total sum of squares:

SSR | sum of squares within samples (error/residual)

+

SSM | sum of squares between samples (treatment/model)

MS = SS/df

sums of squares/ degrees of freedom

o   F larger than 1: more explained than unexplained variation

o   the bigger ‘n – 1’ gets, the larger F gets -> large experiment = high test power

• also for ANOVA: R2 = explained variation / total variation

• ANOVA for an unbalanced design: PERMANOVA

-          the samples are independent random samples,

-          the populations are normally distributed,

-          the population variances are equal.

1.      Independency of replicates

2.      Independency of samples

3.      Normal distribution

4.      Populations have common variance

5.      Additive factor effects

è Shows if there is a difference, but not were exactly

è Compares mean of treatments (explained and unexplained)

nominal: yes or no

interval: counting

if you lose replicates (imbalanced design), rebalance the design (kick out replicates or a whole sample).

With one-factorial ANOVAs, Kruskal-Wallis- and median-tests, one only tests whether or not there are significant differences in a group of means. Post-hoc testing does pairwise (or groupwise) mean comparisons, that indicate which means (or group of means) differ from one another.

Problem:

-          comparing multiple means to one another increases the type I error rate

Solution:

-          the Bonferroni correction:

method to counteract the problem of multiple comparisons

• this of course, increases the type II error!

linear combination of parameters or statistics whose coefficients add up to 0

Regression = best case scenario

> more powerful than ANOVA

> more information content than ANOVA

smoothers = approximating functions that attempt to capture important patterns in the data

Moving average:

Analyse data points by creating series of averages of different subsets of the full data set.

> LOcal regrESSion

Fits simple models to localized subsets of the data to build up a function that describes its variation.

Splits the data into bins and applies polynomial regression to each bin.

Examines the influence of multiple categorical independent variables on one continuous dependent variable.

• combining all treatments of factors with each other

• —> able to investigate interactions btw factors

E.g.: Partial pressure of CO2 and temperature on phytoplankton net growth.

independent variables that affect a dependent variable are uncorrelated

Two factors are crossed when every category of one factor co-occurs in the design with every category of the other factor.

A factor is nested within another factor when each category of the first factor co-occurs with only one category of the other.

Mixed-effect models consider not only fixed (dependent variables) but also random factors (geographic distribution of samples in the field) in your experiment.

Random factors contribute to the unexplained variation. Mixed-effect models avoid that they contribute to the test power of your testing.

-          useful for repeated measurements

-          good in dealing with missing values

Fixed effects: CO2 partial pressure nutrient conc. light regimes predator presence/absence sex

Random effects: genotype plot within region block within experiment split plot within a plot

either crossed or nested

Can handle data that stem from repeated sampling of identical sampling units.

+      reduced unexplained variation

+      reduced amount of sampling units (f.e. organisms)

-          require a further assumption about the data: sphericity

-          special statistical tests needed to analyse data

Most interesting bit in repeated measures: does my effect change over time?

Modelling the relationship between a dependent and more than one independent variables (predictors).

• analysis of covariance

• An ANOVA tests for the influence of one or more categorical independent variables on one dependent variable. Regression analysis tests for the influence of one or more continuous variables on one dependent variable.

• Analysis of covariance (ANCOVA) is a method that combines ANOVA and regression. It measures the influence of one or more categorical predictors and one or more continuous independent variables on one dependent variable. The continuous independent is called covariate and is normally not of interest

the blue area gets divided into the variance explained by the error and the variance explained by the covariate

Estimate the influence of the covariate on the dependent variable.

By incorporating covariates in the model, the amount of unexplained variation is reduced.

⇨     increased test power due to noise reduction

⇨     smaller effects become visible

if your interval data are not normally distributed, you can switch to non-parametric tests (Mann-Whitney U, Wilcoxon)

-> transformation of interval to ranked data

powerful

• Paired tests compare two sets of measurements to assess whether their population means differ. They reduce the amount of unexplained variability in the experiment.

  Therefore, it’s easier to detect differences between treatments, i.e. higher test power.

Presence or absence of snails in two tidal horizons (Yes/No).H0: Proportion of the snails is the same in the two tidal horizons.

• independent samples

Germination of previously cooled or uncooled seeds.H0: There’s no difference in the median of the two populations.

• independent samples

Effect of fertilizer on the canopy height of a crop.H0: There’s no difference in the mean of two populations.

• independent samples

Fouling on structured and unstructured shells.H0: No difference in amount of more fouled structured and more fouled unstructured shells.

• dependent samples

• Number of invertebrate larvae in the middle and at the edges of streams. H0: No median differences between members of pairs of larvae (from middle/edges).

• dependent samples

Fouling on structured and unstructured shells.H0: No difference in fouling rates between structured and unstructured shells.

• dependent samples

sum of squares/ degrees of freedom

want to get significant effect -_> more explained than unexplained variation

correlation is simplest form of interaction, interaction is described by calculating statistics or making graphs.

Correlation quantifies the strength of the linear relationship between a pair of variables, whereas regression expresses the relationship in the form of an equation.

generaly anova:

1. independent observations

2. normaly distributed

3. homogeneity of variances

additionaly:

1. homogeneity of regression slopes

2. linearity

3. covariate is measured without error