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What is statistics? Inference and uncertainty: This is what
statistics is all about.
Statistics consists of a body of methods for collecting and analyzing data. (Agresti & Finlay, 1997)
Developed for interpreting and drawing conclusions from collected data
The major objective of statistics is to make inferences about population from the analysis of the sample data
What does statistics provide?
Design: Planning and carrying out research studies
Description: Summarizing and exploring data
Inference: Making predictions and generalizing about phenomena represented by the data
Population vs. sample
Steps in Planning Statistical analysis
Terms and Terminologies
Population- Total group of samples or individuals that the researcher is interested to study.
Sample- A group of individuals selected from the population
Parameter- is a characteristic of a population
Statistic- is a characteristic of a sample
Variable- characteristic or attribute that can assume different values.
Variate- A random variable taken from a known probability distribution
Terms and Terminologies
Descriptive statistics- describe the relationship between variables. E.g. Frequencies, means, standard deviation
Exploratory statistics- Usually represented in the form of graphs to see the patterns in a datum.
Inferential statistics- are used to draw inferences about a population from a random sample
Terms and Terminologies
Qualitative variable- Also known as categorical variable. Usually measured on a nominal scale.
Quantitative variable- They are measured on a numeric scale. Ordinal, interval and ratio scales are quantitative
Discrete variable- countable in a finite amount of time.
Continuous variable- would (literally) take forever to count. In fact, you would get to forever and never finish counting them
Terms and Terminologies
Random sampling
Systematic sampling
Convenience sampling
Stratified sampling
Cluster sampling
Sampling Error- is the difference between the sample measure and the corresponding population measure
Descriptive vs. Inferential statistics
Descriptive statistics consist of methods for organizing and summarizing information (Weiss, 1999)
Inferential statistics consist of methods for drawing and measuring the reliability of conclusions about population based on information obtained from a sample of the population. (Weiss, 1999)
Types of Statistical Approaches
Descriptive Statistics- Describes your data
- How many?
- How much?
Exploratory Statistics- represented in the form of graphs
- Is there any pattern?
- Are data points clustered or stretched?
Types of Statistical Approaches
Inferential Statistics
- Are there any differences?
- What is the relationship?
- What is the effect?
- Model building
- What determines what?
Distributions
Positively skewed Symmetric
Negatively skewed
Distributions
Distributions
Normal Probability distribution
Mean, Median and mode are same
Bell-shaped curve symmetrical around mean
Probability area under the curve will be 1
Denoted by
Normal Probability Distribution
Areas under a normal distribution curve
Common types of Probability distributions
Other important types of distribution
1.Poisson2.Binomial
Poisson Distribution
used to represent the number of successive independent events of a specified type with low probability of occurrence (< 10%) in some specified interval of time or space.
Example cases of flu
Denoted by
Poisson Distribution
Binomial Distribution
An experiment that consists of n independent, repeated trials, each of which can end in only one of two ways arbitrarily labeled success or failure.
The probability that any trial ends in a successis p (and hence q = 1 p for a failure).
Denoted by
where in
Binomial Distribution
Central Limit Theorem
Sampling distribution of means
As the sample size n increases without limit,the shape of the distribution of the samplemeans taken with replacement from apopulation with mean m and standarddeviation will approach a normaldistribution.
This distribution will have a mean m and astandard deviation /n
Central Limit Theorem
Central Limit Theorem
Importance of Central limit theorem
- we can describe the sampling distribution from any variable without actually having to infinitely sample the population of raw scores.
Types of Variables
Nominal
Ordinal
Interval
Ratio
Types of Variables
Sampling Techniques
Random sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other sampling techniques- Convenience sampling, Sequential sampling, Double sampling and multi-stage sampling
Theory of Probability
Experiment
Outcome
Sample space
Event
Theory of Probability
P [A]= No of Possible outcomes in which an event A occurs
Total No of possible outcomes in the sample space
Where P [A]= Probability that an Event B will occur
P(A)= 0 to 1
P(A)+P(B)+.+P(n)= 1
P(AorB) = P(A)+P(B) >> Disjoint event
P(AandB) = P(A)*P(B) >> Joint eventIndependent events
Theory of probability
P(AUB) = P(A)+P(B)- P(AB) >> contingent joint event
P(AB) = P(A)+P(B)- P(AUB) >> contingent joint event
P(A|B) = P(AB)/P(B) >>conditional probability for A
P(B|A) = P(AB)/P(A) >>conditional probability for B
Definition of ProbabilityA probability measure is a rule, say P, which associateswith each event contained in a sample space S a number suchthat the following properties are satisfied:
1 For any event, A, P(A) 0.2 P(S) = 1 (since S contains all the outcomes, S alwaysoccurs).3 P(not A)+P(A)=1.4 If A and B are mutually exclusive events (that cannot
occur simultaneously) and independent events (that are not linked inany way), then P(A or B) = P(A) + P(B) andP(A and B) = 0
Note: Many elementary probability theorems (rules) follow directlyfrom these definitions.
Confidence Intervals
The range around any hypothetical value of mean () within which 95% of the means of all samples of size n taken from that population will occur.
Denoted by
Where 95% confidence interval for mean, when population variable X is normally distributed and known
Understanding Z-statistic
Confidence Intervals
Distribution of the Z statistic (the ratio of the difference of population mean and sample mean divided by the Standard error of the mean (SEM) obtained by taking the means of a large number of small samples from a normal distribution). The 95% confidence interval obtained by taking the means of a large number of small samples from a normally distributed population with known statistics is indicated by the black horizontal bar enclosed within 1.96 SEM. By chance 95% of the sample means will be within the range 1.96 to +1.96 , with the remaining 5% outside this range
Confidence Intervals
With larger sample sizes, the 95% confidence intervals get smaller
P-Value
It is defined as the probability of getting the observed result, or a more extreme result, if the null hypothesis is true. In other words it is the measure of the likelihood of the result given the null hypothesis is true or the statistical significance of the claim.
range from 0 to 1
P-Value
"P=0.030" is a shorthand way of saying "The probability of getting 17 or fewer male chickens out of 48 total chickens, IF the null hypothesis is true that 50 percent of chickens are male, is 0.030.
It is a usual convention in biology to use
a critical P-value of 0.05 (often called alpha, )
P-Value
This p-value measures how likely it was that you would have gotten your sample results if the null hypothesis were true.
The farther out your test statistic is on the tails of the standard normal distribution, the smaller the p-value will be, and the more evidence you have against the null hypothesis being true
Interpreting P-value
If the p-value is greater than or equal to , you fail to reject Ho.
If the p-value is less than , reject Ho.
p-values on the borderline (very close to ) are treated as marginal results
Interpreting P-value
- Heres how to interpret your results for any given alpha level
To make a proper decision about whether or not to reject Ho, you determine your cutoff probability for your p-value before doing a hypothesis test; this cutoff is called an alpha level ().
Typical values for are 0.05 or 0.01
Interpreting P-value
- How to interpret your results if you use an alpha level of 0.05
If the p-value is less than 0.01 (very small), the results are considered highly statistically significant reject Ho.
If the p-value is between 0.05 and 0.01 (but not close to 0.05), the results are considered statistically significant reject Ho
If the p-value is close to 0.05, the results are considered marginally significant decision could go either way
If the p-value is greater than (but not close to) 0.05, the results are considered non-significant dont reject Ho
Biological vs statistical hypotheses
Biological and statistical hypothesis-
- "Sexual selection by females has not caused male chickens to evolve bigger feet than females
- Male chickens dont have a different average foot size than females
Statistical Hypothesis Statistical Hypothesis- statement about the
probability distribution of populations using one or more data samples
Hypothesis H0: All data samples originate from the same population (or the single data sample is consistent with a given theoretical distribution).
Hypothesis H1: Some data samples do not originate from the same population (or the single data sample is not consistent with the given theoretical distribution).
Statistical Inference and Hypothesis Testing
What do we mean by chance?
What do we mean unlikely?
What do we mean by effect?
Hypothesis and Significance Testing
Hypothesis- is a statement about some characteristic of a variable or a collection of variables. (Agresti & Finlay, 1997).
Significance test- is a way of statistically testing a hypothesis by comparing the data to values predicted by the hypothesis
The Process of Hypothesis Testing
Sample selected at random from very different population may not necessarily be different. Simply by chance the samples from populations 1 and 2 are similar, so you might mistakenly conclude the two populations are also similar
The Mechanism of Hypothesis Testing
The Mechanism of Hypothesis Testing
Even a random sample may not necessarily be a good representative of the population. Two samples have been taken at random from the same population. By chance, sample 1 contains a group of relatively large fish, while those in sample 2 are relatively small.
Type I & Type II errors
Test Statistics and your decision
Type I & Type II errors
Four possible results of hypothesis testing
Parametric statistics
Also known as classical statistics
Parametric tests are designed for analysingdata from a known distribution
ANOVA (1920s and 30s), Multiple Regression (1800s), T-tests (1900s), Pearson Correlation (1880s) are parametric statistical methods
Parametric statistics
General Assumptions of Parametric Statistical Tests
1. The sample of n subjects is randomly selected from the population.
2. The variables are continuous and from the normal distribution
3. The measurement of each variable is based on interval or ratio data
Non parametric Statistics Sometimes called distribution free statistics
Do not require data to be normally distributed
In general, a less powerful test than the analogous parametric test
No normality assumption
Uses less information
Spearmans Rho (1904), Kendalls Tau (1938), Kruskal-Wallis (1950s), Wilcoxon Signed-Ranks Matched Pairs (1940s)
Parametric vs Non Parametric
Parametric test Non-parametric analogT-test (unpaired) Wilcoxon rank sum testPaired t-test Wilcoxon signed rank testANOVA Kruskal-Wallis testRepeated measures ANOVA
Friedman test
The parametric tests are called parametric because, when we calculate the p-value, we use the parameters of the normal distribution: mean and standard deviation
The non-parametric tests do not estimate these parameters, but instead are based on ranks
Hypothesis and Statistical Tests
main goal of a statistical or Hypothesis test-
what is the probability of getting a result like my observed data, if the null hypothesis were
true
Evaluate and compare groups of data
To determine whether hypothesis can be retained or rejected and modified
can refer to a single group
can also refer to two groups
Steps for a hypothesis Test
1. Set up the null and alternative hypotheses: Ho and Ha.
2. Take a random sample of individuals from the population and calculate the sample statistics (means and standard deviations).
3. Convert the sample statistic to a test statistic by
changing it to a standard score (all formulas for test statistics are provided later in this chapter).
4. Find the p-value for your test statistic.
5. Examine your p-value and make your decision.
Structure of Hypothesis Tests
1. Choose the appropriate test.
2. Establish the null and alternate hypotheses.
3. Decide on an acceptable error rate .
4. Compute the test statistic from the data.
5. Compute the p-value.
6. Reject the null hypothesis if p .
Sampling Distributions
Major parametric test statistics -
Z distribution
T distribution
Chi-square
F distribution
Sample size is the key
Sampling test Distributions
Four common probability distributions of sample statistics z, t, chi-square, and F
Z distribution
Represents the probability distribution of a random variable that is the ratio of the difference between a sample statistic and its population value to the standard deviation of the population statistic
Students t Distribution
Chi-square Distribution
represents the probability distribution of a variable that is the square of values from a standard normal distribution
bounded by 0 and infinity
used for interval estimation of population variances
can also be used to determine the probability of obtaining a sample difference (or one smaller or larger) between observed values and those predicted by a model
F Distribution
represents the probability distribution of a variable that is the ratio of two independent chi-square variables, each divided by its df (degrees of freedom) (Hays 1994).
Because variances are distributed as 2, the F distribution is used for testing hypotheses about ratios of variances.
bounded by zero and infinity.
Used to determine the probability of obtaining a sample variance ratio (or one larger) for a specified value of the true ratio between variances
Hypothesis Testing
Null Hypothesis(H)&Alternate Hypothesis(H)
H: = / H: (Two-tailed test)
H: = / H: (one-tailed test)
Types of hypothesis tests
Associations and Differences
Relationship between variables Associations and Differences
Association- The relationship between a wing length and weight of a growing bird
Difference- The relationship between the mean tail length of Gull-billed Tern and the mean tail length of Common Tern.
Difference of mean tests
One sample t-test
Two independent samples t-test
t= SE /n
where t represents the effect size or test statistic
Paired samples t-test
K-independent samples (n>2)
- ANOVA (Analysis of Variance)
One way ANOVA
Two way ANOVA
Difference of mean tests (Non parametric)
- One sample
Runs test
- Two independent samples
Kolmogrov-Smirnov test
Mann Whitney U test
Difference of mean tests (Non parametric)
- Paired samples
Wilcoxon signed Ranks test
Mc Nemars test
Marginal Homogeneity test
- K independent samples
Kruskall- Wallis test
Friedmans Rank test
Test of Proportions, ratios and indices
Chi-square test
Goodness of fit
Correlation
Pearsons product moment correlation (r)
To investigate linear relationships between two independent variables
r -1 to +1
Correlation
Scatter plots with various correlations
Regression
Prediction is made on the assumption the hypothesis is correct
Simple linear regression
Investigate relationships- Dependent and independent variable
Best fit linear line describes relation between X and Y
Regression coefficient/ Coefficient of determination (R)
Regression
Regression lines by gender and parity status for predicting weight at 1 month of age in term babies
Classification of some hypothesis tests
Summary of Statistical Tests
Common Errors of statistical analysis
Samples are not random
Sample size is too low for any meaningful interpretation
Non-independence of sample data
Overuse of non-parametric statistics, even with low sample size
Failure to do a graphical exploration
Common Errors of statistical analysis
Power analysis and effect size
Interpreting simple correlation as cause and effect
Use of complex model and multivariate statistics without verifying the merit of the data
Power of a test
Measure of likelihood of a test reaching a correct conclusion