Attack evenness

Update, September 2024: see the followup article looking at attack evenness in many more leagues and tournaments.

Details

Let’s start with some definitions:

a “lineup” is the 6 players on court (ignoring the libero) along with their positions relative to the setter. Rotation is ignored — when the team rotates on sideout this is still treated as the same lineup, but if the players change position (e.g. the two outsides swap places) that is considered to be a different lineup.
we will calculate attack evenness using a team’s “attack profile”, which is the proportion of attacks that each player in the lineup made. We ignore setter dumps, since these are relatively rare. As an example, let’s say that our opposite made 35% of our attacks, one outside hitter made 25% and the other 30%, and the two middles made 5% of attacks each. That attack profile would be represented as [0.35 0.25 0.30 0.05 0.05].
given an attack profile, we will calculate its evenness by comparing it to a “reference profile” representing an perfectly even attack profile, so we need to define what this is. A team in a given rotation has four attack options: the front-row middle, two outsides, and the opposite. A perfectly even distribution would be when each of those four players make 25% of the attacks in that rotation. When we average over all 6 rotations, and assuming that the middles are being replaced by the libero in back court (and thus only front row for 3 out of every 6 rotations), the overall attack profile is [0.25 0.25 0.25 0.125 0.125] (i.e. the opposite and outsides each make 25% of the total attacks, and the middles make 12.5%).

Note that we do not necessarily expect any team to actually exhibit exactly this perfectly even attack profile in practice, nor are we saying that this attack profile is optimal. It simply provides us with a common reference against which we can compare all teams.

Now, given an attack profile $\bm{a}$ and the reference profile $\bm{r}$ , attack evenness is calculated by taking the difference of the two:

$aev = 1 - 0.5 * \sum_{i=1}^{5}{|a_i - r_i|}$

where we are summing over the five attackers. The resulting value can be interpreted as follows: an aev value of 1 represents a perfectly even attack distribution. The more uneven the distribution, the lower the aev value will be. The difference of the aev from 1 is the proportion of attacks that would have needed to go to a different attacker in order to have achieved a perfectly even distribution (e.g. for an aev of 0.9, re-distributing 10% of attacks in an optimal way would have allowed that team to achieve perfectly even distribution).

To give some examples:

our perfectly even attack profile of [0.25 0.25 0.25 0.125 0.125] has an evenness of 1
an attack profile of [0 0 0 0 1] (i.e. all attacks made by one of the middle hitters, which is the most uneven it can be) has an evenness of 0.125
an attack profile of [0.2 0.5 0.2 0.05 0.05] (one of the outsides dominates, making half of all attacks) has an evenness of 0.75
an attack profile of [0.2 0.3 0.25 0.15 0.1] (a reasonably even profile) has an evenness of 0.925

So the theoretical range of values is 0.125 to 1, but realistically we are probably expecting to see values in the 0.75 to 0.95 range.

A team will play with a variety of lineups over time, and so to get the team’s overall evenness we calculate the evenness separately for each lineup in each match and average the results, weighting by the number of attacks made by each. We can also collate various performance metrics as we go, and see how those relate to evenness. Lineup/match combinations that only made a small number of attacks (< 10 by that lineup in the match) are discarded, since their attack profiles will be more likely to appear unbalanced simply by virtue of a small sample size.

Example

Let’s explore this concept on data from the 2023/24 men’s German Bundesliga competition.

Team	Ladder position	aev	aev SD
SVG Lüneburg	4	0.879	0.062
BERLIN RECYCLING Volleys	1	0.868	0.067
VfB Friedrichshafen	3	0.857	0.062
WWK Volleys Herrsching	5	0.856	0.062
TSV Haching München	11	0.848	0.069
HELIOS GRIZZLYS Giesen	2	0.846	0.063
SWD powervolleys DÜREN	6	0.845	0.064
Baden Volleys SSC Karlsruhe	8	0.839	0.084
VC Bitterfeld-Wolfen	7	0.837	0.073
FT 1844 Freiburg	10	0.833	0.075
ASV Dachau	9	0.806	0.074
Energiequelle Netzhoppers KW-Bestensee	12	0.801	0.073

The Ladder position column gives the team’s position on the league ladder at the end of the main competition round (i.e. excluding finals). Lüneburg and Berlin had the highest attack evenness, and Dachau and Netzhoppers thel lowest. We can see a rough correspondence between ladder position and attack evenness, with higher-ranked teams having higher evenness. München and Giesen are exceptions to this, with München having higher evenness than ladder position, and Giesen the opposite.

The aev SD column in the above table gives the variability in evenness across a team’s different lineups. If we plot this:

There is a negative relationship: the higher a team’s evenness, the more consistent they tend to be (lower SD). But there is considerable variability. Karlsruhe had conspicuously higher variability than other teams. The four teams that joined the first-division league in 2023/24 (Karlsruhe, Freiburg, Dachau, and Bitterfeld-Wolfen) had four of the five highest variability values.

What about variability between setters? Most teams use multiple setters — how does a team change when the setter changes?

Team	Setter	N	aev	aev SD
ASV Dachau	Luca Russelmann	539	0.858	0.073
ASV Dachau	Moritz Gärtner	1053	0.780	0.059
BERLIN RECYCLING Volleys	Johannes Tille	1670	0.866	0.069
BERLIN RECYCLING Volleys	Leon Dervisaj	586	0.875	0.060
Baden Volleys SSC Karlsruhe	Milan Kvrzic	296	0.778	0.092
Baden Volleys SSC Karlsruhe	Tobias Hosch	1419	0.852	0.076
Energiequelle Netzhoppers KW-Bestensee	Djifa Julien Amedegnato	1186	0.801	0.074
Energiequelle Netzhoppers KW-Bestensee	Jonas Lind	565	0.801	0.070
FT 1844 Freiburg	Fabian Hosch	1175	0.843	0.067
FT 1844 Freiburg	Lorenz Rudolf	407	0.804	0.087
HELIOS GRIZZLYS Giesen	Fedor Ivanov	1944	0.844	0.064
HELIOS GRIZZLYS Giesen	Jan Röling	166	0.865	0.050
SVG Lüneburg	Hannes Gerken	424	0.875	0.063
SVG Lüneburg	Maxwell David Elgert	1420	0.880	0.061
SWD powervolleys DÜREN	Christopher Gavlas	277	0.858	0.066
SWD powervolleys DÜREN	Leo Meyer	1652	0.843	0.064
TSV Haching München	Eric Paduretu	889	0.852	0.075
TSV Haching München	Marcell Mikuláss Koch	973	0.845	0.064
VC Bitterfeld-Wolfen	Benedikt Gerken	263	0.799	0.074
VC Bitterfeld-Wolfen	Matus Jalovecky	1809	0.843	0.071
VfB Friedrichshafen	Aleksa Batak	2296	0.860	0.060
WWK Volleys Herrsching	Eric Burggräf	1652	0.863	0.054
WWK Volleys Herrsching	Severin Brandt	201	0.797	0.087

(Not all setters are necessarily present in this table. To reduce the effects of lineups that only played a small number of rallies, this table only includes lineups that made at least 10 attacks in a match, and setters who made at least 100 sets to such lineups.)

Some teams (e.g. Berlin, Düren, Lüneburg, München, Netzhoppers) show similar evenness between their setters, but others (notably Dachau, Herrsching, and Karlsruhe) are quite different.

In general, where a team has two setters in this table they either have similar evenness values to each other, or the setter with the higher N (their preferred setter) has the higher evenness. Differences here could reflect a number of factors — for example, setters that are routinely brought on court in double-substitution situations will spend more time playing with a different opposite to the other setter. But differences in evenness might also be reflective of differences in setter skill in delivering balls to their attackers.

Implications

Do we see any relationships between attack evenness and rally or match outcomes? In the first table above we can see a rough correspondence between a team’s attack evenness and their ladder position at the end of the main competition round. Let’s look at rally win rate against attack evenness (both averaged over a team’s lineups, as described above).

We see a strong positive trend: higher evenness values are associated with higher rally win rates. The variation around the trend is potentially interesting: München (who placed low on the league ladder) had a noticeably lower rally win rate than might be expected given their attack evenness, whereas Giesen (who placed second on the ladder) was the opposite with a rally win rate well above the trend line.

Mechanisms — how does evenness help?

This relationship between rally win rate and attack evenness could potentially reflect a number of factors:

higher evenness represents a more diversified set of attack options, which allows the team to exploit advantageous matchups as they arise during matches. It is also more robust to variations in the performance of individual attackers. If someone is having an off day, other attackers can perform well to compensate. This allows for better average performance over time. A team with uneven attack might not have sufficient depth to compensate if the main attackers are not delivering.
A more diversified attack also presents greater difficulties for the defenders, and so (all other things being equal) would give higher rally win rates.
increases in attack evenness generally correspond to higher usage of the middle attackers (highly uneven attacks typically rely heavily on one or more of the outside hitters or opposite, so an increase in evenness will generally correspond to more middle attack). The middle attack also has the highest kill rates (for evidence, see e.g. our league leaderboard for that season in which 9 of the top 10 attackers by kill rate were middle hitters). So an increased evenness will naturally bring higher rally win rates by virtue of using more first-tempo attack.
it’s also possible that teams can only play with high attack evenness if they have sufficient depth to their attacking roster, and the increased rally win rate is just because they are stronger, not specifically because they are playing more evenly.

To try and make some sense of this, we fit some statistical models (binomial generalized additive models, with rally win rate as the response variable). All models include lineup and opposition team as predictors. These are factors, meaning that an average rally win rate is estimated for each lineup/opposition team combination. To some models we also add terms that allow the rally win rate to vary as a smooth function of evenness and/or middle usage (how much attacking the middles do).

A model with terms for lineup and opposition team is substantially improved by adding the evenness term (change in AIC of -12.4 for those that want to know). In other words, after accounting for the relative strength of a lineup and its opposition team, we still find that evenness has a positive relationship with rally win rate. This gives us some confidence to rule out point (d) above.
The same base model with terms for lineup and opposition team is not noticeably improved by adding the middle term (change in AIC of 1.3). Similarly, the model with terms for lineup, opposition team, and evenness is not noticeably improved by adding the middle term (change in AIC of 0.2). So middle usage does not help to explain variations in rally win rate.

This not to say that there is no benefit to playing more with the middles. Different lineups will use their middles to different degrees, so the lineup term already soaks up some of the variation in rally win rate due to variations in middle usage. What this result is saying is that we don’t get a better fit to the data by explicitly accounting for middle usage on top of that. But we do get a better fit when we add the evenness term. This lends credence to point (c) above: increasing evenness will increase rally rate due to increased use of middles, but that does not explain all of the variability in rally win rate. There is more to the story — increased rally win rate with increased evenness is not simply because we use the middles more.
Finally, we model the attack kill rate of just the non-middle attackers in each lineup. A model with terms for lineup and opposition team is substantially improved by adding the evenness term (change in AIC of -11.3). This is saying that the kill rate of non-middle attackers is improved when attack evenness is increased. This provides further evidence that the increased rally win rate is not simply because we are using our middles more. It is also consistent points (a) and (b) above. Having a greater number of genuine attack options makes life harder for the defenders, so kill rates go up for outsides and opposites, who should be facing less well-formed blocks and less well-positioned defenders.

So attack evenness helps, not just by allowing the setter to use more middle attack, but by diversifying the overall attack portfolio.

To finish our examination of the mechanisms, let’s look at the evenness term in the first model above:

The grey band shows the uncertainty in the model fit, and it becomes wider at the ends because there are fewer data points at extreme evenness values (the tick marks on the x-axis show the distribution of evenness values). The mean fit (the black line) slopes upwards for the most part, showing that the rally win rate increases with increasing evenness, as we expect. To put some context on the size of this effect, the upwards part of the curve has a slope of about 2.5 (increasing from -0.5 at evenness 0.6 to 0 at evenness of 0.8). An increase of 5% evenness (from, say, 0.75 to 0.8) would increase the “partial effect” by 2.5 * 0.05 = 0.125. This is a logistic regression model, so the “partial effect” is on a logit scale and we can calculate the odds ratio as exp(0.125) = 1.13. That means that a 5% increase in evenness increases my odds of winning the rally by about 13%.

The curve shows an interesting flattening and downturn for very high evenness values. This suggests that beyond a certain level (about 0.9), further increases in evenness are associated with slightly decreased rally win rates. However, this is not straightforward to interpret. Generalized additive models can be sensitive to outliers near the ends of their range, so this downturn might be reflecting some anomalously-low win rates by a small number of lineups with high evenness values, rather than indicating a more pervasive relationship in the data. But it is also plausible that it is a real effect. In order to achieve an extremely even attack profile, the setter must by definition spread the attack across all attackers, which increases the likelihood of giving at least some balls to attackers who are not performing well, or who are in difficult situations.

Opportunity effect

How does passing performance affect evenness? The natural expectation is that better passing gives the setter more options, and should therefore correspond to higher attack evenness. Certainly, better passing offers more opportunities to set first-tempo attacks (or slide attacks, for women), which as we have seen above will increase evenness. But in reality a setter needs to accommodate variations across attacker strength, blocker matchups, and many other factors. Setters also use different strategies that will affect distribution, and so the overall relationship between passing and evenness is likely to be a mixture of many things.

In this plot we use attack evenness on reception-phase attacks only (i.e. the first attack a team makes after receiving), since this should be most strongly affected by pass quality.

We see the expected positive relationship, suggesting that better passing does indeed offer the chance for higher attack evenness — but again with some variation, which might reflect influences from those setter-strategy factors mentioned above.

Addendum: player load

As a side-effect of the above analyses, we can also compile a table of an individual player’s attack load (i.e. the proportion of a team’s attacks that the player made while on court) along with their attack performance metrics. The plot below shows the player load on the x-axis (relative to the role average, so a value of 0.05 for an outside hitter means that individual made 5% more attacks than the average outside hitter). The y-axis shows attack efficiency, also relative to the role average. Only players who made at least 50 attacks are shown.

Players lying on the right side of the plot (higher than average load) might be considered to be important attackers for their teams. Those in the upper half (higher than average performance) are the better performers. The better performing players — not surprisingly — tend to be on stronger teams (bluer-coloured points).

Players in the top-right quadrant are heavily used and also performing well (e.g. Mote, Ahyi, and House). Players in the top-left quadrant might be viewed as under-utilized relative to their performance — perhaps because they play on teams with other, stronger attackers.