Statistical Analysis ANOVA Roderick Graham Fashion Institute of Technology

ANOVA  ANOVA means Analysis Of Variance.  With ANOVA, we have one variable that is ungrouped, and one categorical, grouped variable.  The question that we try to answer with ANOVA is: “for any collection of groups, is at least one group different from the others?”  This is hypothesis testing. This time your critical statistic will be an F ratio.

The types of procedures we’ve used… Regression X = ungrouped/scale Y = ungrouped/scale Chi- Square X = grouped/category Y = grouped/category ANOVA X = grouped/category Y = ungrouped/scale

Steps in Conducting ANOVA tests  Step 1 – State the null hypothesis and critical region  Step 2 – Identify the critical statistic  Step 3 – Compute the test (calculated) statistic  Step 4 – Interpret results

Example: Capital Punishment and Religious Affiliation  Suppose we ask 20 people to rank their support for capital punishment on a scale from 1 to 30.  1 being absolutely no support under no circumstances  30 being always support under any circumstance  Suppose these 20 people were equally divided into 5 religions (Protestant, Catholic, Jewish, Buddhist, and other)  Now, we have an ungrouped, scale variable (support for capital punishment) and a grouped, categorical variable (religion). This is the perfect situation for ANOVA!

Example: Capital Punishment and Religious Affiliation  Now let’s say we got this data:  What do we do? ProtestantCatholicJewishBuddhistOther

Example: Capital Punishment and Religious Affiliation Step 1 – State the null hypothesis and critical region H 0 : “The population means for each category are the same.” H 1 : “At least one of the population means is different.” Let’s set our critical region at.05 (Meaning we will accept 95% of all findings, and if we get a calculated statistic that falls in the.05 region, we will reject the null hypothesis).

Example: Capital Punishment and Religious Affiliation Step 2 – Identify the critical statistic.  To get the critical statistic, (F critical), we must look for two values.  dfb (degrees of freedom between) – columns for F chart  dfb = (k – 1), with k = number of categories  dfw (degrees of freedom within) – rows for F chart  dfw = (N – k), with N = number of respondents, and k = number of categories

Example: Capital Punishment and Religious Affiliation Step 2 – Identify the critical statistic. For our example we have a k of 5 (five categories) and an N of 20 (20 respondents). Thus, dfb = (5 – 1) = 4 dfw = (20 – 5) = 15, F(critical) = 3.06

Example: Capital Punishment and Religious Affiliation What the upcoming symbols mean:  SST = sum of squares total  SSW = sum of squares within  SSB = sum of squares between  MSB = mean squares between  MSW = mean squares within

Example: Capital Punishment and Religious Affiliation Step 3 – Compute the test/calculated statistic requires using these equations. Important…these formulas should be used in this order!!!

Example: Capital Punishment and Religious Affiliation  Let’s take a closer look at the beginning formulas Note: SSB will have to be calculated for each category and then summed. In our example, we have five religious groups, so we will compute SSB five times, and then sum. Mean value for the entire sample Number of cases in EACH category Total number of cases in a sample Mean value of EACH category

Example: Capital Punishment and Religious Affiliation  Understanding the final formula. It is a ratio of the differences between groups and the differences within groups  When MSB and MSW are similar, the F is low, and the less chance we will reject the null.  But when these two values differ, the F increases. We then begin to believe that at least one of the populations that these samples represent is different from the other populations.

Example: Capital Punishment and Religious Affiliation  This is all we need to being building our table for ANOVA.  What new information do we need in order to use our starting formulas? ProtestantCatholicJewishBuddhistOther

Example: Capital Punishment and Religious Affiliation  Setting up the table for ANOVA…  You also need:  The N (sample), = 20, and N k (no. of cases per category) = 4  The mean of the entire sample, = ( )/20 = 16.6 ProtestantCatholicJewishBuddhistOther xx2x2 xx2x2 xx2x2 xx2x2 xx2x ∑ = 16.6

Our F(critical) was….3.06 Our F(test) was…2.57 Thus, we do not reject the null hypothesis.

Your turn…  Are sexually active teens better informed about AIDS than teenagers who are sexually inactive? A 15 item test of knowledge about sex was administered to teens who were “inactive”, “active with one partner” and “active with more than one partner”. Here are the results. Test at the.05 level. Give H 0, H 1, F-critical, F-ratio, Concluding Results InactiveActive – 1 Partner Active – More than 1 Partner

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