<p>Alpha compares the total test variance to the sum of the item variances (* the number of items/number of items -1) .  Total test variance is just the sum of the item variances and covariances.  As you can see from your correlation matrix, you have some negatively correlated items.  This will reduce the total variance to be less than the sum of the item variances.  In your case, the total test variance is 6.197 and the sum of the item variances is 8.015 with k =5 and thus</p>

<pre><code>alpha = (6.198 - 8.015)/6.198  * 5/4 =  -.366




lowerCor(data)
        rMEQQ01 rMEQQ02 rMEQQ03 rMEQQ04 rMEQQ05
rMEQQ01  1.00                                  
rMEQQ02 -0.26    1.00                          
rMEQQ03  0.29   -0.02    1.00                  
rMEQQ04 -0.15    0.09   -0.02    1.00          
rMEQQ05 -0.45    0.30   -0.32    0.20    1.00 
</code></pre>

<p>For a better understanding of alpha and the omega statistics, I recommend you read our paper in press at Psychological Assessment: Revelle and Condon, From alpha to Omega, a tutorial (2020) preprint at <a href="https://psyarxiv.com/2y3w9/" rel="nofollow noreferrer">https://psyarxiv.com/2y3w9/</a>.  </p>
Why is the value of alpha different between psych::omega and psych::alpha in R?
<p>I have data from a five-item measure completed by about 260 participants and estimated internal consistency with the psych library in R. I got a value of -0.36 (I triple-checked for reverse-coded questions). My data doesn't cover a broad range of the possible scores -- due to the nature of the group I am sampling from -- so this is a plausible number. But the scale I'm using is well validated, admittedly with low typical values of alpha ranging from 0.5--0.7. </p>

<p>When looking for alternative methods to assess my data I used psych::omega. This returns an alpha value of 0.57 alongwith the omega statistic I was looking for. This alpha value is much more in keeping with other literature which uses the scale in question. I have also looked at other validated measures from my study and the alpha values from psych::alpha and psych::omega are equal. It is only on this particular measure which is known to have low internal consistency where the difference appears.</p>

<p>My question: why are the alpha values different between psych::alpha and psych::omega?</p>

<p>The code below includes the actual data and will spit a warning about negatively correlated items. I've checked and rechecked my score-coding and it's correct so I am confident we can ignore this warning.</p>

<pre><code>library("psych")

data &lt;- structure(list(rMEQQ01 = c(2, 3, 3, 3, 3, 3, 3, 1, 5, 3, 3, 3, 
3, 3, 3, 3, 3, 5, 3, 2, 5, 2, 3, 3, 3, 3, 3, 2, 3, 2, 2, 1, 3, 
5, 3, 4, 3, 4, 5, 4, 3, 3, 2, 5, 4, 2, 3, 3, 5, 3, 3, 5, 5, 4, 
2, 2, 4, 1, 4, 1, 2, 1, 5, 2, 4, 3, 3, 1, 3, 3, 4, 2, 4, 2, 2, 
3, 5, 4, 3, 3, 2, 5, 3, 5, 1, 5, 3, 5, 1, 3, 3, 3, 3, 3, 5, 3, 
5, 5, 2, 5, 2, 5, 3, 5, 3, 3, 5, 4, 3, 3, 4, 3, 3, 4, 3, 2, 3, 
4, 3, 2, 3, 5, 5, 3, 4, 4, 3, 5, 3, 2, 3, 2, 5, 3, 3, 5, 3, 3, 
3, 2, 5, 5, 4, 4, 5, 4, 1, 3, 3, 3, 4, 3, 3, 3, 3, 3, 2, 4, 3, 
3, 3, 4, 4, 3, 3, 3, 4, 3, 1, 2, 3, 3, 5, 4, 3, 4, 2, 2, 3, 4, 
2, 2, 1, 4, 3, 5, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 4, 5, 2, 3, 5, 
4, 3, 4, 4, 4, 3, 3, 3, 5, 2, 3, 3, 5, 5, 4, 2, 3, 3, 3, 3, 3, 
3, 3, 2, 3, 1, 3, 2, 1, 2, 5, 3, 5, 1, 3, 3, 2, 3, 4, 5, 3, 3, 
3, 4, 3, 1, 3, 3, 3, 3, 5, 1, 3, 4, 5, 3, 4, 3, 3), rMEQQ02 = c(2, 
2, 2, 1, 2, 1, 2, 3, 2, 2, 3, 2, 1, 1, 2, 1, 2, 1, 2, 3, 4, 2, 
1, 1, 3, 2, 3, 2, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 
1, 1, 3, 2, 2, 1, 3, 3, 2, 4, 1, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 
3, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 1, 2, 1, 3, 3, 3, 1, 1, 2, 2, 
1, 3, 2, 3, 2, 2, 2, 2, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 
1, 2, 1, 3, 1, 2, 2, 2, 3, 3, 2, 2, 3, 1, 2, 2, 2, 3, 1, 1, 2, 
1, 1, 2, 3, 1, 1, 3, 3, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 1, 
3, 2, 3, 1, 3, 3, 3, 3, 3, 1, 2, 2, 1, 1, 2, 1, 2, 2, 3, 3, 2, 
3, 1, 2, 3, 1, 3, 2, 3, 2, 2, 2, 2, 3, 2, 3, 3, 1, 2, 1, 2, 2, 
2, 3, 2, 2, 2, 2, 4, 1, 2, 2, 1, 2, 2, 1, 3, 2, 2, 1, 3, 1, 3, 
2, 3, 1, 2, 1, 3, 1, 2, 2, 3, 3, 2, 3, 2, 2, 1, 2, 2, 3, 2, 1, 
1, 1, 2, 3, 2, 2, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 3, 3, 1, 3, 
1, 2, 2, 2, 1, 1, 1), rMEQQ03 = c(1, 3, 1, 3, 3, 2, 2, 5, 4, 
3, 3, 3, 4, 3, 3, 3, 3, 5, 3, 2, 5, 1, 3, 3, 3, 3, 2, 3, 3, 1, 
2, 2, 3, 1, 3, 4, 3, 1, 5, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 
1, 4, 3, 1, 3, 3, 1, 3, 3, 1, 2, 3, 3, 4, 3, 2, 3, 2, 2, 2, 3, 
3, 3, 3, 3, 5, 3, 2, 3, 3, 1, 1, 1, 1, 3, 3, 3, 2, 3, 1, 4, 3, 
3, 5, 3, 5, 1, 3, 5, 1, 1, 2, 1, 2, 3, 3, 3, 3, 3, 5, 3, 3, 3, 
2, 2, 3, 5, 1, 1, 1, 3, 2, 3, 3, 3, 3, 3, 1, 3, 3, 4, 3, 3, 2, 
5, 1, 3, 5, 1, 1, 3, 4, 2, 4, 3, 3, 1, 1, 3, 5, 1, 3, 3, 3, 3, 
1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 3, 3, 1, 3, 2, 3, 5, 4, 4, 4, 1, 
2, 3, 3, 3, 2, 1, 3, 2, 3, 2, 2, 1, 3, 3, 1, 3, 5, 4, 3, 3, 1, 
3, 3, 1, 2, 3, 1, 4, 5, 1, 2, 3, 5, 2, 5, 3, 5, 5, 4, 2, 3, 3, 
3, 2, 4, 3, 3, 2, 2, 1, 2, 2, 2, 2, 5, 5, 3, 3, 2, 3, 3, 2, 3, 
4, 3, 5, 3, 3, 3, 5, 5, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5), 
    rMEQQ04 = c(3, 2, 3, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 
    2, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 2, 4, 3, 3, 3, 
    3, 3, 1, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 5, 3, 3, 3, 3, 
    5, 3, 3, 3, 5, 5, 4, 1, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 
    3, 4, 3, 3, 2, 3, 3, 3, 2, 2, 3, 3, 3, 1, 3, 2, 1, 3, 2, 
    2, 3, 3, 2, 3, 3, 1, 3, 1, 4, 3, 3, 1, 3, 3, 3, 5, 3, 3, 
    3, 3, 2, 3, 3, 3, 2, 3, 4, 2, 3, 3, 3, 2, 2, 3, 3, 3, 3, 
    3, 3, 3, 4, 3, 3, 5, 1, 1, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 
    3, 3, 2, 3, 2, 3, 3, 2, 3, 4, 5, 3, 3, 2, 3, 3, 3, 3, 3, 
    3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 
    3, 3, 1, 3, 1, 2, 2, 1, 2, 3, 3, 3, 5, 4, 1, 1, 1, 3, 5, 
    4, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 2, 
    3, 3, 4, 3, 4, 3, 3, 1, 2, 3, 3, 4, 3, 4, 3, 3, 3, 3, 5, 
    3, 3, 3, 3, 2, 3, 3, 3, 1, 4, 3, 3, 3, 3, 3, 3, 2), rMEQQ05 = c(4, 
    0, 4, 2, 6, 0, 2, 0, 2, 2, 0, 4, 2, 4, 4, 0, 0, 4, 2, 2, 
    0, 4, 4, 0, 4, 2, 6, 4, 2, 2, 4, 6, 2, 0, 2, 0, 0, 2, 0, 
    2, 2, 6, 0, 0, 0, 0, 0, 4, 0, 4, 4, 0, 4, 0, 4, 4, 0, 6, 
    2, 6, 4, 6, 2, 6, 2, 2, 2, 4, 6, 4, 2, 6, 2, 6, 6, 2, 0, 
    0, 4, 2, 6, 6, 4, 0, 4, 0, 6, 4, 0, 2, 2, 2, 2, 4, 0, 2, 
    0, 0, 4, 0, 4, 2, 0, 0, 2, 2, 0, 0, 4, 6, 2, 4, 0, 2, 4, 
    4, 0, 2, 6, 2, 4, 0, 0, 4, 0, 4, 2, 4, 2, 0, 4, 0, 0, 4, 
    2, 0, 2, 2, 0, 0, 0, 4, 0, 4, 0, 4, 2, 6, 6, 0, 0, 4, 4, 
    6, 2, 2, 4, 0, 0, 0, 2, 0, 2, 2, 2, 4, 2, 2, 6, 4, 2, 4, 
    0, 0, 0, 0, 2, 0, 2, 0, 4, 4, 4, 4, 4, 2, 6, 4, 0, 4, 0, 
    2, 0, 0, 0, 6, 2, 0, 6, 4, 2, 2, 0, 2, 0, 2, 6, 2, 2, 0, 
    6, 2, 2, 0, 0, 0, 4, 0, 4, 2, 6, 2, 4, 2, 4, 2, 4, 6, 6, 
    6, 4, 0, 0, 0, 4, 6, 2, 6, 6, 2, 2, 0, 0, 2, 2, 6, 2, 2, 
    6, 6, 0, 0, 4, 2, 4, 4, 4, 0, 2, 0)), row.names = c(NA, -260L
), class = "data.frame")

alpha &lt;- psych::alpha(data) # alpha$total$raw_alpha = -0.3665156
omega &lt;- psych::omega(data) # omega$alpha = 0.5710576
```


</code></pre>