Statistics Final Exam

Any study with more than one independent variable (factor)

ESP:

• Economic
• Scientific
• Practical

1. Economical: Can look at several IVs simultaneously

2. Scientific: Allows us to examine:

• -Effects of each IV on the DV
• -Effect of the COMBINATION of the IVs on the DV

3. Practical: More likely publishable

A matrix represents the "cell means" for each group

• 
• M= 10 2 8 10

We call this a 2x2 design because there are two IVs, each with 2 levels

3 X 2 = 2 IVs 1 w/ 3 levels, 1 w/ 2

• 3 x 4 x 2 = 3 IVs
• 1 w/ 3 levels, 1 w/ 4 levels, 1 w/ 2 levels

Two-way ANOVA

Three-way ANOVA

• Statistical concerns
• -more variables
• -less power
• -need more subjects
• Practical

• -Tests for the effects of each independent variable separately
• -And the effects from combining independent variables (interaction)

M of 1 level of an IV averaged across 1 level of another IV.

• 6
• 6 9

*Average cells for each IV

• M of Therapy A = 9
• M of Therapy B = 6
• M of Drug 1 = 6
• M of Drug 2 = 9

There is a significant effect of each IV = Main Effect for both Drug Type and Therapy Type

• Different levels of different independent variables
• 

The effects of Drug 1 are best when combined w/ Therapy B

• When the effects of an IV are different at the different levels of another IV
• - the combined (joint) effects of a pair of IVs on the DV

• -Does not mean that one IV depends on (influences) the other IV
• -They are still independent on one another (drug type can not influence therapy type)

• -Words
• -Numbers: the differences in cell means across one row (or column) will be different than the other differences in cell means across another
• -Graphs: when a departure from parellilism exists, a significant interaction might also exist

• The effect of Therapy Type X is different for different types of drugs.
• 

• Compare the differences (subtract) in cell means from one row to another...
• Therapy A: 10 - 8 = 2
• Therapy B: 2 -10 = -8

If the pattern of differences is not the same across rows, then there might be an interaction...

• Technically, the correct way to display a factorial design is with a bar graph...
• -If the bars from the two levels of an IV aren't parallel across the levels of the other IV, there is an interaction

• Typically, people present a line graph
• It doesn't matter which variable that's on the X axis

• 1. Only 1 Main Effect is significant
• 2. Both Main Effects, but not the Interaction
• 3. Both Main Effects AND the Interaction
• 4. Only the Interaction is Significant
• 5. Nothing Significant

10 - 10 = 0, 5 - 5 = 10

10 - 5 = 5, 20 - 15 = 5

• Chart
• 

• 10 - 1 = 9
• 1 - 10 = -9

• Be very cautious in interpreting the main effects if a study has a significant Interaction
• -sometimes the main effects are important above and beyond the interaction
• -sometimes not

But the interaction doesn't cross over

Interaction effects magnitude of results, but not overall pattrn

• Can't interpret the main effects without understanding the interaction...
• 

The interaction is driving the data

• A situation in the factorial ANOVA in which a combination of variables has an affect that could not be predicted from the effects of the two variables individually
• i.e. combination of sensitivity and test difficulty

Factorial research design in analysis of variance with two variables that each divide the groups.

Analysis of variance for a two-way factorial design?

A variable that separates groups in analysis of variance.

Analysis of variance in which there is only one grouping variable.

• The difference between groups on one grouping variable ina factorial design in analysis of variance; result for a grouping variable, averaging across the levels of the other grouping variable(s)
• i.e. one for sensitivity and one for test difficulty

Its the number inside the box, duh

The means of each particular level. M of cell 1 + cell 2. M of cell 1 + cell 3. Etc. and so forth.

The impact of one grouping variable depends on the level of another grouping variable.

The pattern of differences in cell means across one row will not be the same as the patterns of differences in cell means across another row.

Whenever there is an interaction, the pattern of bars on one section of the graph is different from teh pattern on the other section of the graph

-A study can have any combinatin of interaction and main effects.

In this result, the main effect for arousal holds up over and above the interaction. The effect for arousal is there for both easy and hard tasks; in both cases, low arousal produces the least performance, moderate the next most, and high arousal the most. -There is still an interaction because how much high arousal produces better performance than moderate arousal is more for hard than for easy tasks.

An association between scores on two variables.

A relation between two variables tht shows up on a scatter diagram as the dots roughly following a straight line.

A relations between two variables that shows up ona scatter diagram as dots following a systematic pattern that is not a straight line.

If you plotted income and shoe size- random dots everywhere.

• A positive slope. A relation between two variables in which high schores on one go with high scores on the other, mediums with mediums, and lows with lows.
• On a scatter diagram, the dots roughly follow a straght line sloping and to the right.

• A negative slope. A relation between two variables in which high scores on one go with low scores on the other, mediums with mediums, and lows with highs.
• On a scatter diagram, the dots roughly follow a straight line sloping down and to the right.

How much there is a clear pattern of some particular relationship between two variables.

Looking at a scatter diagram gives you a rough understanding of the relationship between two variables. A correlation coefficient gives you the exact number that determines the direction and strength of this relationship.

• Need to determine what is a high score and what is a low score. + deviation = raw score above the mean. - deviation = raw score below the mean.
• X - M_x
• Y - M_y

When you multiply the deviation score of one variable with that of another. If the product is positive (+ x + = + ; - x - = +) then the correlation is positive. Vice versa.

Add them up, if they are close to zero, then there is not a linear correlation.

The larger the number, the stronger the correlation. But this can also be misleading because how large is large? Hopefully very large. Bigger is always better.

• + = positive correlation
• - = negative correlation

...straightforward enough for you, eh?

• 1. It gets larger with more people.
• 2. It gets larger as the scores for each variable have more variation.

[(SS_x)(SS_Y)]^1/2 Divde the sum of the product of deviation scores by this correction number.

The result of dividing the sum of the products of deviation scores by the correction number.

• + or : tells you the direction
• 0 - 1 : tells you the strength of the correlation

Σ[(X – Mx) * (Y- My)]

• r = Σ[(X – Mx) * (Y- My)]
• √(SSx * SSy)

*The correction number controls for N and SD

Is it significantly different than zero?

Null = In the population the true relation between the two variable is no correlation (r = 0)

• t = r
• √[(1-r2) /(N – 2)]

df = N - 2

The path of causal effect:

If X is thought to cause Y, then the direction of causality is from X to Y

• 1. X could be causing Y
• 2. Y could be causing X
• 3. Z could be causing both X and Y

• Correlational research design- any design other than a true experiment.
• A correlational research design is not necessarily statistically analyzed using the correlation coefficient, and some studies using the experimental research designs are most appropriately analyzed using a correlation coefficient.

Square the correlations = r²

This is the proportionate reduction in error.

• Like age in our lab.
• Situation in which you figure your corrleation but only a limited range of the possible values on one of the variables is included in the group studied.

One of the reasons why the dots may not fall close to the line is inaccurate measurement.

They fuck up most statistical analysis.

The correlation coefficient only works if the relationship is linear. This is why it is important to use a scatterplot before you calculate Pearson's r.

The equivalent of a correlation coefficient for rank-ordered scores.

• 1. Outliers
• 2. Non-linear patterns
• 3. Restrictions in range
• 4. Inaccurate measurements

Variable that is used to predict scores of individuals on other variables

A variable that is predicted

Formula for predicting a person's score on a criterion variable based on the person's score on one or more predictor variables

Particular fixed number added into the prediction.

• Number multiplied by a person's score on a predictore variable as part of a linear prediction rule.
• The slope of the regression line.

Y = a + (b)(X)

• Y: person's predicted score on the criterion variable
• X: person's score on the predictor variable
• a: regression constant
• b: regression coefficient

• The difference between a prediction rule's predicted score on the criterion variable and a person's actual score on the criterion variable is called error. Take each error and square it.
• Minimizes distance between actual data and predicted values.

Sum of the squared differences between each predicted score and actual score on the criterion variable.

a = MY – (b)(MX)

• Normality
• Homoscedasticity
• Relationship is linear

• Experimental: manipulation (only of IV) ; control (all other variables constant)
• Correlational: anything that is not a true experiment; when we don't directly manipulate the IV

• b = Σ[(X – Mx) * (Y – MY)]
• SSx

• To see if X predicts Y (relationship between these variables)
• Testing to compare slopes across studies

• Regression coefficient in standard deviation units.
• It shows the predicted amount of change in standard deviation units of the criterion variable if the value of the predictor variable increase by one standard deviation.

β = b * (√SSX /√SSY)

They are the same.

• β acts as r so can be used on t distribution.
• t test for the correlation applies to both types of regression coefficients.

The hypothesis test for a regression coefficient (for both b and β) tests whether the regression coefficient is significantly different from 0.

• Hypothesis-testing procedures used when the varaibles of interest are nominal variables.
• Comparing an observed frequency distribution to an expected frequency distribution.

Examines how well an observed frequency distribution of a nominal variable fits some expected pattern of frequencies.

Examines whether the distribution of frequencies over the categoreies of one nominal variable is unrelated to the distribution of frequencies over the categories of a second nominal variable.

df = N_categories - 1

X² = E [(Observed - Expected)2]/Expected

• 1. Outliers
• 2. Unreliable Measurement
• 3. Reliability
• 4. Heteroscedasticity

When we have one nominal variable

When we have > one nominal variable.

• Proportion of smart/dumb people will be equal (even if we dont know waht those proportions will be)
• The two variables will be independent

[Total for that row / total # of subjects ] * total for that column

df = (# of rows - 1) * (# of columns - 1)

The proportion of subjects with a particular combination of traits.

• Phi Coefficient
• Cramer's

• 1. Independence: each observation is independent of all other observations
• 2. Minimum expected frequency size: depends on who you ask
• 3. NOT NORMALITY: this is a nonparametric test

• 1. Run with the data as is
• 2. Delete the case
• 3. Transform the data
• 4. Do something else (nonparametric)

• + Square Root Transformation: positively skewed data
• ++ Logarithm Transformation
• +++ Inverse Transformation
• - Reflect scores

NONPARAMETRIC

• Dep t test ----------- Wilcoxon Sign Rank Test
• Ind t test ------------- Mann Whitney U Test
• One way ANOVA -- Kruskal Whallis Test
• Pearson's r ----------- Spearman's rank order

• ≈ High and low scores
• Medians will be ≈

Used to analyze ordinal data

• Goodness of Fit
• Null: the distribution of peoples across categories will be the same.
• Test for Independence
• Null: two populations are the same. proportions for both methods are the same.

Itself.