The Scientific Method Formulation of an H ypothesis P lanning an experiment to objectively test the hypothesis Careful observation and collection of D ata from the experiment I nterpretation of the experimental results

Steps in Experimentation H Definition of the problem Statement of objectives P Selection of treatments Selection of experimental material Selection of experimental design Selection of the unit for observation and the number of replications Control of the effects of the adjacent units on each other Consideration of data to be collected Outlining statistical analysis and summarization of results D Conducting the experiment I Analyzing data and interpreting results Preparation of a complete, readable, and correct report

The Well-Planned Experiment Simplicity –don’t attempt to do too much –write out the objectives, listed in order of priority Degree of precision –appropriate design –sufficient replication Absence of systematic error Range of validity of conclusions –well-defined reference population –repeat the experiment in time and space –a factorial set of treatments also increases the range Calculation of degree of uncertainty

Types of variables Continuous –can take on any value within a range (height, yield, etc.) –measurements are approximate –often normally distributed Discrete –only certain values are possible (e.g., counts, scores) –not normally distributed, but means may be Categorical –qualitative; no natural order –often called classification variables –generally interested in frequencies of individuals in each class –binomial and multinomial distributions are common

Terminology experiment treatment factor levels variable experimental unit (plot) replications sampling unit block experimental error  planned inquiry  procedure whose effect will be measured  class of related treatments  states of a factor  measurable characteristic of a plot  unit to which a treatment is applied  experimental units that receive the same treatment  part of experimental unit that is measured  group of homogeneous experimental units  variation among experimental units that are treated alike

Barley Yield Trial Experiment Hypothesis Treatment Factor Levels Variable Experimental Unit Replication Block Sampling Unit Error

Hypothesis Testing H 0 :  = ɵ H A :   ɵ or H 0 :  1 =  2 H A :  1   2 If the observed (i.e., calculated) test statistic is greater than the critical value, reject H 0 If the observed test statistic is less than the critical value, fail to reject H 0 The concept of a rejection region (e.g.  = 0.05) is not favored by some statisticians It may be more informative to: –Report the p-value for the observed test statistic –Report confidence intervals for treatment means

Hypothesis testing It is necessary to define a rejection region to determine the power of a test Correct  Type I error  Type II error 1 -  Power Decision Accept H 0 Reject H 0 Reality H 0 is true  1 =  2 H A is true  1   2

Power of the test Power is greater when –differences among treaments are large –alpha is large –standard errors are small

Review - Corrected Sum of Squares Definition formula Computational formula –common in older textbooks correction factor uncorrected sum of squares

Review of t tests To test the hypothesis that the mean of a single population is equal to some value: where df = n-1 df = 6 df = 3 df =  Compare to critical t for n-1 df for a given  (0.05 in this graph)

Review of t tests To compare the mean of two populations with equal variances and equal sample sizes: where df = 2(n-1) The pooled s 2 should be a weighted average of the two samples

Review of t tests To compare the mean of two populations with equal variances and unequal sample sizes: where df = (n 1 -1) + (n 2 -1) The pooled s 2 should be a weighted average of the two samples

Review of t tests When observations are paired, it may be beneficial to use a paired t test – for example, feeding rations given to animals from the same litter t 2 = F in a Completely Randomized Design (CRD) when there are only two treatment levels Paired t 2 = F in a RBD (Randomized Complete Block Design) with two treatment levels

Measures of Variation s (standard deviation) CV (coefficient of variation) se (standard error of a mean) LSD (Least Significant Difference between means) L (Confidence Interval for a difference between means) L (Confidence Interval for a mean) (standard error of a difference between means)