ANOVA Statistical Programming with PHP

ANOVA Statistical Programming with PHP
Pages: 1, 2, 3, 4

The bulk of the code involves calculating the value of various instance variables to use in subsequent reporting steps. Most of these instance variables are associative arrays with indices such as total, between, and within. This is because the ANOVA procedure involves computing the total variance (in our test scores) and partitioning it into between-group (i.e., between treatment levels) and within-group (i.e., within a treatment level) variance estimates.

At the end of the analyze method we evaluate the probability of the observed F score by first instantiating an FDistribution class with our degrees of freedom parameters:

$F = new FDistribution($this->df["between"], $this->df["within"]);

To obtain the probability of the obtained F score we subtract 1 minus the value returned by the cumulative distribution function applied to the obtained F score:

$this->p = 1 - $F->CDF($this->f);

Finally, we invoke the inverse cumulative distribution function using 1 minus our alpha setting (i.e., 1 - 0.05) in order set a critical F value that defines the decision criterion we will use to reject the null hypothesis which states that there is no difference between treatment-level means.

$this-crit = $F->inverseCDF(1 - $this->alpha);

If our observed F score is visibly greater than the critical F score, we can conclude that at least one of the means differs significantly from the others. A p value (i.e., $this->p) value less than 0.05 (or whatever your null rejection setting is) would also lead you to reject the null hypothesis.

The formula for decomposing the total sum of squares (first term) into a between-groups component (second term) and a within-group component (third term) appears in Figure 1.

$\sum_{t=1}^{k}\sum_{i=1}^{n_t}(y_{ti} - \bar{y})^2 = \sum_{t=1}^{k}n_t(\bar{y}_t - \bar{y})^2 + \sum_{t=1}^{k}\sum_{i=1}^{n_t}(y_{ti} - \bar{y}_t)^2$
Figure 1. Formula for decomposing the sum of squares.

The symbol stands for the grand mean and the symbol stands for the treatment mean.

Step 2: Show Raw Data

It is always good to begin your analysis by making sure that you've properly loaded your data. We can call the showRawData() method to dump our test-anxiety data table to a web browser.

<?php
/*
* Output contents of database table.
*/  
function showRawData() {
  global $db;
  $data        = $db->tableInfo($this->table, DB_TABLEINFO_ORDER);
  $columns     = array_keys($data["order"]);
  $num_columns = count($columns);

  ?>

  <table cellspacing='0' cellpadding='0'>
    <tr>
      <td>
        <table border='1' cellspacing='0' cellpadding='3'>
        <?php
          print "<tr bgcolor='ffffcc'>";

          for ($i=0; $i < $num_columns; $i++) {
            print "<td align='center'><b>".$columns[$i]."</b></td>";
          }

          print "</tr>";

          $fields = implode(",", $columns); 
          $sql    = " SELECT $fields FROM $this->table ";
          $result = $db->query($sql);

          if (DB::isError($result)) {
            die( $result->getMessage());
          } else {
            while ($row = $result->fetchRow()) { 
              print "<tr>";

              foreach( $row as $key=>$value) {
                print "<td>$value</td>";
              }

              print "</tr>";
            }
          }
          ?>
        </table>
      </td>
    </tr>
  </table>
  <?php
}  
?>

This code generates as output the table below:

Table 1. Show Raw Data

id	anxiety	score
1	low	26
2	low	34
3	low	46
4	low	48
5	low	42
6	low	49
7	low	74
8	low	61
9	low	51
10	low	53
11	moderate	51
12	moderate	50
13	moderate	33
14	moderate	28
15	moderate	47
16	moderate	50
17	moderate	48
18	moderate	60
19	moderate	71
20	moderate	42
21	high	52
22	high	64
23	high	39
24	high	54
25	high	58
26	high	53
27	high	77
28	high	56
29	high	63
30	high	59

A tip for data miners: Maybe you already have some data in your databases to which you can adapt this code. Look for situations where you have an enum data type to act as your treatment-level field and a corresponding integer or float column that measures some response associated with that treatment-level.

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