Biostatistics For Animal Science and Statistics General Knowledge

What are the steps of planning and conducting an experiment?

• Introduction to the problem

• Statement of the hypothesis

• Description of the experimental design

• Collection of data

• Analysis of the data

• Interpretation of the results relative to the hypotheses

What is the purpose of an experimental design?

It declares how the data is obtained

What is a statistical model?

Follows the experimental design and is explained by a math formula and includes three components

What are the three components included in a statistical model?

1. Definitions of means (expectations)

2. Definition of dispersion (variances and covariances)

3. Definition of distribution

Experimental Unit

the unit which a treatment is applied to

Sample Unit

what the effect of treatment is measured on

Repetitions

repeated measures on the same experimental unit and independent of one another

What is the measure of experimental error in regression or one-way ANOVA?

Residual Mean Square

Residual Mean Square

average of the squared differences between actual observed values and model predicted values

What are the two types of experimental error?

Systemic and Random

Systemic Error

Consistent effects which change measurements under study and can be assigned to some source

• produce bias in estimation

• can come from lack of uniformity in conducting the experiment

What are some examples of systemic error?

• Use of uncalibrated instruments

• Unaccounted temperature effects

• Biases in using instruments

Random Error

Occurs due to random, unpredictable phenomena and produce variability that can’t be explained

• has an expectation of 0, over a series of replicates it will cancel out

What type of error can always be found in biological material?

Random Error

What are examples of random error?

• Genetic makeup of animals

• Measurement error

• The degree to which measurements are rounded

Accuracy

Indicates how close to an aim, often represented by how close the estimated mean of replicated measures is to the true mean, the closer the more accurate

Precision

How close together trials are to each other; represented by how close together the measurements are regardless of how close they are to the true mean, explains repeatability of the results

What type of error affects the precision of an experiment?

Random Errors

What type of error affects the accuracy of an experiment?

Systemic Errors

What are the characteristics of an effective experimental design?

1. Yields unbiased results

2. Has high power and low likelihood of type I error

3. Representative of the population to which results will apply

What does the power of the experiment depend on?

the number of replicates and degrees of freedom

What is needed to determine the number of replications?

• estimation of variance

• the effect size of practical importance and which should be found significant

• the power of test (1-B) or probability of finding a significant effect of a given size

• the significance level (a), the probability of a type I error

• the type of statistical test

Blocking

grouping together homogenous units to account for known sources of variability

What is the purpose of blocking?

Experimental error decreases and the possibility of finding a difference between treatments increases

What is an example of blocking?

Breeds of pigs defined as blocks and pigs within each breed are fed different feeds

Randomized Complete Block Design

1. Experimental units are divided to alpha treatments and b blocks, each treatment appears in each block only once

2. The treatments are assigned to units in each block randomly

What does the model of a randomized complete block design include?

• In the end, it equals an observation in treatment and block

• Sum of:

  • The overall mean

  • The effect of treatment

  • The effect of block

  • Random error with mean and variance

  • The number of treatments

  • The number of blocks

Factorial experiment

has two or more sets of treatments that are analyzed at the same time, all factor levels are tests

Main Effect

the effect of a factor alone

Interaction

the effect of different factors acting together

Characteristics of experimental design of a factorial experiment

the experimental design is completely randomized and combinations of factors are applied to experimental units

In a factorial experiment what hypothesis test should be carried out first?

The hypothesis test for the interaction must be carried out first, if the effect of the interaction isn’t significant the main effects are tested. If the interaction is significant, the tests for the main effects are meaningless.

−2 log likelihood (−2LL):

In binary logistic regression, −2LL is analogous to the sum of squared residuals in a multiple linear regression; the larger the absolute value of −2LL, the worse the agreement between predicted probabilities and actual group membership. LL values are computed to compare fits for different binary logistic regression models. Multiplying an LL value by −2 converts it to a chi-square distributed variable. Critical values of the chi-square distribution can be used to assess the statistical significance of one model or the significance of change in goodness of fit when variables are added to a model. See also log likelihood (LL).

a:

In a mediated causal model, an a coefficient corresponds to the path from X1 (the initial cause) to M (the hypothesized mediating variable). For purposes of significance tests, the raw-score regression coefficient is used; in diagrams to represent the path model, the standardized version (beta coefficient) usually appears.

Adjusted means:

In an analysis of covariance, adjusted Y means are created by subtracting the part of the scores that are related to or predictable from the covariates; this adjustment may correct for any confound of the covariates with treatment and/or suppress some of the error variance or noise and increase the statistical power of tests of group differences.

Aroian test:

A z test for H0: ab = 0 that uses a different method to estimate SEab from the one proposed by Sobel (1982).

Autoregressive model:

A model for serial dependence in a time series such that X at Time t is predictable from X at Time t − 1 and possibly X at earlier times such as t − 2 and t − 3. It is called “auto” because values of X are predicted from values of X.

B:

The coefficient for a predictor variable (X) in a logistic regression model. For a one-unit increase in X, there is a B-unit change in the log odds ratio. The value of B is not easy to interpret directly; instead, B is used to calculate other values that are more directly interpretable, such as the odds ratio and the probability of membership in each group.

b:

In a mediated causal model, the b coefficient corresponds to the path from M (the mediating variable) to Y (the final outcome variable). For purposes of significance tests, the raw-score regression coefficient is used; in diagrams to represent the path model, the standardized version (beta coefficient) usually appears.

Backward method of entry:

A form of statistical multiple regression that begins with all candidate predictor variables included in the equation; in each step, the predictor variable for which the removal leads to the smallest decrease in the R2 for the model is dropped. The process of variable deletion ends when dropping a variable would result in a significant decrease in the total R2.

Bentler comparative fit index (CFI):

A goodness-of-fit index based on information that compares model fit for the specified model with model fit for the null model the (which represents an assumption that none of the variables are related). The CFI is downwardly adjusted as the number of free parameters is increased. It has a range from 0 to 1.00; values > .95 are generally viewed as indicating adequate model fit.

Bootstrapping:

An empirical method for estimation of standard errors and confidence intervals, used in situations where there is no simple formula to calculate these, or where violations of assumptions (such as multivariate normality) make estimating these problematic. It is a resampling method. In effect, the program draws many samples from the data set (if number of bootstraps = 2,000, then it draws 2,000 samples) and calculates all the model parameters for each sample. An empirical distribution of the 2,000 estimates can then be set up to summarize how each parameter (such as the a × b product for an indirect path in the mediation analysis example) varies across 2,000 samples. This empirical sampling distribution is used to estimate SE and confidence intervals.

C (level of confidence):

When setting up a confidence interval, this is usually arbitrarily set at 95% or 90%. If all assumptions for use of confidence intervals are correct, then in the long run, if we set up thousands of confidence intervals using samples from the same population, C% of the confidence intervals are expected to contain μ; (1 – C%) are expected not to contain μ.

c:

In a mediation analysis, c represents the “total” relationship between the initial cause X1 and the outcome Y.

c´:

In a mediation analysis, c´ represents the direct path from X1 to Y, when the indirect path from X1 to Y (via the mediating variable M) is controlled for in the analysis.

Canonical correlation (rc):

The correlation between scores on a discriminant function and group membership. This is analogous to an η coefficient in an ANOVA. The term canonical implies that there are multiple variables on one side (or both sides) of the model. The canonical correlation, rc, for each discriminant function is related to the eigenvalue for each function.

Causal model:

A causal model is often represented by a path diagram; unidirectional arrows represent hypotheses that variables are causally related, and bidirectional arrows represent hypotheses that variables are not causally associated. These models are sometimes used to help data analysts decide what partial correlation and regression analyses to perform on data obtained from nonexperimental research. The term causal model is potentially misleading; we cannot make causal inferences about variables on the basis of nonexperimental data. A causal model is purely theoretical; it represents hypotheses about the ways in which two or more variables may be related to one another, and these hypothesized relations may include both causal and noncausal associations.

Causal path:

In a causal model, a unidirectional arrow that points from X toward Y denotes the theoretical existence of a causal connection in which X causes or influences Y (X → Y).

Centering:

Scores on a predictor variable are centered by subtracting the sample mean from the scores on each variable. When both predictors are quantitative, scores on both X1 and X2 should be centered before creating a product term, X1 × X2, to represent an X1-by-X2 interaction. (That is, X1–centered = X1 – sample mean of X1.) Centering reduces the correlation between the product term (X1 and X2) and its components, X1 and X2; centering does not change the statistical significance or effect size for the interaction, but it can make X1 and X2 appear to be stronger predictors of Y.

Centroid:

A multivariate mean; in discriminant analysis, the centroid of a group corresponds to the mean values of the scores on D1, D2, and any other discriminant functions. In a graph, the centroid is the midpoint of the cluster of data points in the space defined by axes that represent the discriminant functions.

Change score:

In a simple pretest–posttest design in which the same behavior or attitude is measured twice, a change score (sometimes denoted d for “difference”) is computed by subtracting the pretest score from the posttest score: d = Ypost − Ypre. The null hypothesis of interest is either H0: d = 0 (for studies of a single group) or H0: d1 = d2 = ··· = dk (for comparisons of treatment impact across k groups).

Chi-square goodness-of-fit index

One of several indexes of fit that describes how well the overall model paths and coefficients reproduce the variance/covariance information in the data. A large χ2 indicates poor model fit. For large-N samples, χ2 tends to be statistically significant even for reasonably good fit.

Chi-square test:

For the overall logistic regression model, the chi-square test tests the null hypothesis that the overall model including all predictor variables is not predictive of group membership. Typically, the value of chi-square is obtained by taking the difference between the −2LL value for the model that includes all predictor variables of interest and the −2LL value for a model that includes only a constant. Using binary logistic regression, a data analyst usually hopes for a large χ2 as evidence that the predictor variables are useful in predicting likelihood of group membership.

Classification errors:

When the predicted group membership for a case (on the basis of the values of discriminant function scores or a binary logistic regression) is not the same as the actual group membership for that case, this is referred to as a classification error.

Classification table:

To assess how well a logistic regression or a discriminant analysis predicts group membership, a contingency table can be set up to show how well or how poorly the predicted group memberships correspond to the actual group memberships.

Cohen’s d:

A unit-free effect size that is not related to N, sample size. It describes a difference between means in numbers of standard deviations. Cohen’s d = (M – μhyp)/SD for the one-sample t test. Cohen’s d = (M1 − M2)/SDpooled for the independent-samples t test.

Cohen’s kappa:

A statistic that assesses the degree of agreement in the assignment of categories made by two judges or observers (correcting for chance levels of agreement).

Communality:

A communality, sometimes denoted h2, is an estimate of the proportion of variance in each of the original p variables that is reproduced by a set of retained components or factors. Communality is obtained for each variable by summing and squaring the correlations (or loadings) for that variable across all the retained factors.

Confirmation bias in publication:

The tendency for studies with results that are not statistically significant to remain unpublished.

Confirmatory factor analysis (CFA):

A model that represents indicator variables with paths (loadings) that relate them to latent variables (factors). Unlike exploratory factor analysis, a confirmatory factor analysis model is usually highly restricted. For example, the number of factors is fixed, and often each indicator variable is related to only one latent variable.

Constant-only model:

See null model.

Constraints on SEM model parameters:

There are many ways model parameters can be constrained. One type of constraint is created when there is no direct path between two variables. (The absence of a path is equivalent to constraining the path coefficient between these variables to be 0.) Another constraint occurs when a fixed numerical value is specified ahead of time for a path coefficient. Another occurs when two or more path coefficients are set to equal each other.

Construct validity:

The degree to which an X variable really measures the construct that it is supposed to measure. In practice, construct validity can be assessed by examining whether scores on X have the sizes and signs of correlations with other variables (which may include experimental manipulations) that they should have, according to theories about the nature of the variables and their interrelationships.

Content validity:

The degree to which the content of questions in a self-report measure covers the entire domain of material that should be included (based on theory or assessments by experts).

Control variable:

See covariate

Convergent validity:

The degree to which a new measure, X´, correlates with an existing measure, X, that is supposed to measure the same construct.

Correctly specified model:

A model is correctly specified if it includes all the variables that should be taken into account (e.g., all variables that are confounded with or causally related to X1 or Y, or all variables that interact with X1 or Y) and, also, if it does not include any variable that should not be included. A well-developed theory can be helpful in deciding what variables need to be included in a model; however, we can never be certain that the models we use to try to explain relations among variables in nonexperimental studies are correctly specified. Thus, we can never be certain that we have taken all potential rival explanatory variables into account; also, we can never be certain that our statistical analysis provides accurate information about the nature of the relation between X1 and Y.

Correlated-samples t test:

See paired-samples t test.

Covariate:

In analysis of covariance or a multiple regression, a covariate is a variable that is included so that its association with the Y outcome and its confound with other predictors can be statistically controlled when assessing the predictive importance of other variables that are of greater interest. Covariates are often quantitative variables (but may be dummy variables).

Cox and Snell’s R2:

This is one of the pseudo-R2 values provided by SPSS as an overall index of the strength of prediction for a binary logistic regression model.

Cronbach’s alpha (α):

An index of internal consistency reliability that assesses the degree to which responses are consistent across a set of multiple measures of the same construct, usually self-report items.

Data-driven regression:

In data-driven regression, variables are entered on the basis of statistical criteria (e.g., in each step, the variable that increases R2 the most is entered, only if its F ratio exceeds a preset criterion).

Determinant of a matrix:

The determinant of a sum of cross products, or correlation matrix, is a single-number summary of the variance for a set of variables when intercorrelations among variables are taken into account.

Diagonal matrix:

A matrix in which all the off-diagonal elements are 0.

Difference score:

See change score.

Dimension reduction analysis:

When a discriminant analysis yields several discriminant functions, the data analyst may think of each discriminant analysis as a “dimension” along which groups differ. The discriminant functions are rank ordered in terms of their strength of association with group membership, and sometimes, only the first one or two discriminant functions provide useful information about the prediction of group membership. When a researcher looks at significance tests for sets of discriminant functions and makes the judgment to drop some of the weaker discriminant functions from the interpretation and retain only the first few discriminant functions, it is called a dimension reduction analysis. The descriptions of differences among groups are then described only in terms of their differences on the dimensions that correspond to the retained discriminant functions.

Dimensionality:

The dimensionality of a set of X variables involves the following question: How many different constructs do these X variables seem to measure? The most common way of assessing dimensionality is to do factor analysis or principal components analysis and to decide how many factors or components are needed to reproduce the correlation matrix R adequately. If these factors or components are meaningful, they may be interpreted as dimensions, latent variables, or measures of different constructs (such as different types of mental ability or different personality traits).

Direct-difference t test:

See paired-samples t test.

Direct effect:

This is denoted c´ in a mediated path model; it represents any additional predictive association between X1 and Y, after controlling for or taking into account the mediated or indirect path from X1 to Y via the mediating variable (M or X2).

Discriminant function:

A weighted linear combination of scores on discriminating variables; the weights are calculated so that the discriminant function has the maximum possible between-groups variance and the minimum possible within-groups variance. For convenience, discriminant function scores are generally scaled so that, like z scores, they have variance equal to 1.0.

Discriminant function coefficients:

These are the weights given to individual predictor variables when we compute scores on discriminant functions. Like regression coefficients, these discriminant function coefficients can be obtained for either a raw score or standard (z) score version of the model; the coefficients that are applied to standardized scores are generally more interpretable.

Discriminant validity:

When our theories tell us that a measure X should be unrelated to other variables such as Y, a correlation of near 0 is taken as evidence of discriminant validity, that is, evidence that X does not measure things it should not be measuring.

Disordinal:

In a disordinal interaction, the lines for a graph of group means cross. This tells us that the rank order of A means differ across levels of the B factor.

Dummy-coded dummy variable:

Dummy coding involves treating each dummy variable as a yes/no question about group membership; a “yes” response is coded 1 and a “no” response is coded 0. In dummy coding of dummy variables, members of the last group receive codes of 0 on all the dummy variables.

Effect:

In analysis of variance, this term has a specific technical meaning. The “effect” of a particular treatment, the treatment given to Group i, is estimated by the mean score on the outcome variable for that group (Mi) minus the grand mean (MY). The effect for Group i is usually denoted by αi; this should not be confused with other uses of alpha (alpha is used elsewhere to refer to the risk for Type I error in significance test procedures and to Cronbach’s alpha reliability coefficient). The “effect” αi represents the “impact” of the treatment or participant characteristics for Group i. When α1 = +9.5, for example, this tells us that the mean for Group 1 was 9.5 points higher than the grand mean. In the context of a well-controlled experiment, this may be interpreted as the magnitude of treatment effect. In studies that involve comparisons of naturally occurring groups, the effect term is not interpreted causally but rather as descriptive information about the difference between each group mean and the grand mean.

Effect-coded dummy variable:

Effect coding involves treating each dummy variable as a yes/no question about group membership; a “yes” response is coded 1, and a “no” response is coded 0. However, the last group (the group that does not correspond to a yes answer on any of the effect-coded dummy variables) receives a code of –1 on all the dummy variables.

Eigenvalue (λ):

In the context of discriminant analysis, an eigenvalue is a value associated with each discriminant function; for each discriminant function, Di, its corresponding eigenvalue, λi, equals SSbetween/SSwithin. When discriminant functions are being calculated, the goal is to find functions that have the maximum possible values of l, that is, discriminant functions that have the maximum possible between-group variance and the minimum possible within-group variance. In the context of factor analysis or principal-components analysis, an eigenvalue is a constant value that is associated with one of the factors in a factor analysis (or one of the components in a principal-components analysis). The eigenvalue is equal to the sum of the squared loadings on a factor (or on a component). When the eigenvalue for a factor or component is divided by k, where k is the number of variables included in the factor analysis, it yields the proportion of the total variance in the data that is accounted for or reproducible from the associated factor or component.

Empirical keying:

A method of scale construction in which items are selected for inclusion in the scale because they have high correlations with the criterion of interest (e.g., on the Minnesota Multiphasic Personality Inventory, items for the depression scale were selected because they predicted a clinical diagnosis of depression). Items selected in this way may not have face validity; that is, it may not be apparent to the test taker that the item has anything to do with depression.

Equivalent models:

Models are equivalent when they have identical goodness of fit but include paths that correspond to different hypotheses about which variables are causally related, or direction of causality.

Error variance:

The proportion of variance in scores for a dependent variable that is not predictable from independent variables included in the analysis.

Error variance suppression:

When we obtain a substantially smaller SSresidual term by including a covariate in our analysis (as in Equation 8.1), we can say that the covariate Xc acts as a “noise suppressor” or an “error variance suppressor.”

exp(B):

The exponential of B, which can also be represented as eB. This is used in binary logistic regression.

Experimental control:

Researchers may use a variety of strategies to “control” variables other than the manipulated independent variable, which could influence their results. For example, consider the possible influence of gender as a third variable in the study of the effects of caffeine on anxiety. Women and men may differ in baseline anxiety, and they may differ in the way they respond to caffeine. To get rid of or control for gender effects, a researcher could hold gender constant, that is, conduct the study only on men (or only on women). Limiting the participants to just one gender has a major drawback: It limits the generalizability of the findings (the results of a study run only with male participants may not be generalizable to women). The researcher might also include gender as a variable in the study; for example, the researcher might assign 10 women and 10 men to each group in the study (the caffeine group and the no caffeine group).

Exploratory factor analysis:

An exploratory factor analysis initially estimates k loadings for k measured variables on k factors. In later steps, a reduced number of factors are examined to assess how well they can reconstruct the correlation matrix among the variables. In contrast to confirmatory factor analysis, exploratory factor analysis does not set constraints a priori on the number of factors or which variables load on which factors. (SPSS permits the user to specify the number of factors to be retained.)

Exponential function:

The exponential function applied to a specific numerical value such as a is simply ea (i.e., the exponential of a is given by the mathematical constant e raised to the ath power). It is sometimes written as exp(a).

Extraction:

This is a computation that involves estimating a set of factor or component loadings; loadings are correlations between actual measured variables and latent factors or components.

Face validity:

The degree to which it is obvious what attitudes or abilities a test measures from the content of the questions posed. High face validity may be desirable in some contexts (when respondents need to feel that the survey is relevant to their concerns), but low face validity may be preferable in other situations (when it is desirable to keep respondents from knowing what is being measured to avoid faking, social desirability bias, or deception).

Factor (1 Definition)

In the context of factor analysis (Chapter 12), a factor is a latent or an imaginary variable that can be used to reconstruct the observed correlations among measured X variables. By looking at the subset of X variables that have the highest correlations with a factor, the researcher decides what underlying dimension or construct might correspond to that factor. For example, if a set of mental ability test scores are factor analyzed, and the first factor has high positive loadings (positive correlations) with the scores on the variables vocabulary, reading comprehension, analogies, and anagrams, the researcher might decide to label this factor “verbal ability.”

Factor (2 Definition)

In the context of analysis of variance, a categorical predictor variable is usually called a factor. In an experiment, the levels of a factor typically correspond to different types of treatment, or different dosage levels of the same treatment, administered to participants by the researcher. In nonexperimental studies, the levels of a factor can correspond to different naturally occurring groups, such as political party or religious affiliation.

Factor score:

A score for each individual participant that is calculated by applying the factor score coefficients for each factor to z scores on all items.