I find this an intriguing problem - how do you compare three related, yet separate factors in their influence on an essentially subjective (military effectiveness) outcome. You can find facts and figures to go with all three of these factors however it is unlikely that any of them will have a relationship with one another. The most you can observe is the outcomes - a military force that is successful or not, while the underlying factors and the working of the mechanism of the system remain obscure.
The aim of this blog is to record interesting facts or sources as I come across them which might be relevant, so I shall add to it as I go along and open it up to comments for readers to join in with good ideas.
The starting point is to agree a definition for Military Effectiveness, in wartime this is winning battles, operations, campaigns and finally the war, with an acceptable level of casualties and general destruction. However there is a tension here which is well explained by Jamel Ostwald's Vauban Under Siege: Engineering Efficiency and Martial Vigor in the War of the Spanish Succession (Brill, 2007) when he describes Vauban's system of siege-craft as an efficiency paradigm designed to reduce casualties however which took time and so only allowed three sieges a year, with the result that campaigns became indecisive. He contrasts this with the approach of later generals - a "Cult of Vigour", which speeded up the process at the cost of higher casualties yet allowed campaigns a chance of obtaining a decisive result.
This balance of casualties against overall results has a great relevant for twentieth century conflicts such as the Great War and particularly for the Soviet-German War, where the Soviet ability to launch multiple offensives simultaneously stretched German strategic reserves beyond breaking point. In the First Period (June 1941 to November 1942) the Red Army was only able to launch single offensives and often with long gaps between them. By the Second Period (November 1942 to January 1944) several offensives could be launched at the same time, for instance Operations Mars & Uranus, though there could still be lengthy gaps between offensives. It was in the Third Period (January 1944 to May 1945) when the Red Army could launch multiple offensives across the front, with one following the other with hardly a break that stressed the Ostheer and led to major defeats such as the destruction of Heeresgruppe Mitte. Tempo of operations was a key factor in keeping the advance moving and not allowing the frontline to solidify into heavily fortified entrenchments.
One approach might be to find one or more comparison armies to give a 'benchmark' against which the Red Army could be assessed. What we are looking for is an army that uses railways, horse-drawn wagons and limited motor transport, relies on infantry and heavy artillery for breakthroughs, tanks for exploitation, airpower used to support the battlefield and a limited level of technology. Italy and Hungary might be candidates from their level of GDP/head however Soviet forces defeated them consistently in the field.
The obvious one is Germany itself yet the Heer had probably the highest level of military effectiveness against which even the wealthy and technologically advanced British and American forces struggled to prevail. The Waffen SS had high levels of equipment yet suffered high casualties (without the need for draconian discipline,) despite being a highly effective military force. By contrast, the Luftwaffe Field Divisions had an equally talented manpower pool (the personnel would all have been regarded as NCO material by the Heer,) a level of equipment on a par with the Heer and a lamentable operational record, so there was nothing inherent about being German that conferred military effectiveness, rather it was the long standing traditions of the Heer and its operating philosophy that achieved effectiveness. This was something that the Red Army lacked, it was an army that expanded hugely both before and during the war while being based on modest foundations.
A less obvious candidate might be the British Army of the Great War, since it had many of the characteristics of the Red Army and it went through a similar period of rapid expansion in personnel, officers and equipment, drawing on a population with little military experience and it went through a similar learning curve of military efficiency, having to devise new tactics and weaponry suitable for the war it was engaged in. It suffered heavy casualties in 1916 and 1917 gaining the tactical skills that would raise its efficiency in 1918 to the successes won in the Hundred Days of 1918, ironically when it was suffering manpower shortages in much the same way as the Red Army would experience in 1944. Both used infantry and heavy artillery, supported by heavy tanks to break the Germans line and then lighter tanks to exploit into the depths. I did initially think the French Army might be a candidate, especially given that faulty doctrine cost it eye watering casualties in both 1914 and 1917 however it did not expand from a limited base to such a degree and had a wider professional base to draw on.
This table is taken from Walter Scott Dunn's Stalin's Keys to Victory: The Rebirth of the Red Army with the data drawn from G. F. Krivosheev Soviet Casualties and Combat Losses in the Twentieth Century which Dunn uses to show that the ratio of Killed to Wounded fell steadily over the course of the war which he claims demonstrates the rising tactical efficiency of the Red Army. The overall percentage of casualties (red line) does fall over the period with an average of 17.5% for the dataset while the ratio of killed to wounded does fall significantly from 1943 onwards, with an average of 0.53 for the dataset.