Do vacuum tube test values reflect sound quality?

As lewn stated, one cannot tell by numbers what a tube will sound like.  The sound will be determined by the particular amp and the particular brand and construction of the tube for any given type.  Depending on the application, a tube can test quite poor, but can sound perfectly good in an amp that does not push the tube very hard.  It may also sound like a sibling tube even if the two test very differently.

I notice you are using very expensive Telefunken ECC801S input tubes.  I hope you got it from reliable sources because the Chinese have become quite good at making fakes, including fakes with the raised Telefunken diamond in the bottom of the glass envelope.  I got my pair for my phono stage a long time ago, and even then I worried about fakes.  The boxes they came in looked very clean and bright, which worried me until I tried to open them and they started to fall apart from age.  That was a good sign to me.    

@lewm @larryi Thank you for the replies.

The Telefunken came with my LTA which I bought used.  The seller seems to be knowledgeable about tubes.  He paid $375 for the Telefunken tubes.  He's a straight shooter, so I believe that to be correct.

But, as far as my knowledge, I'm no expert with tubes.  I'm learning.  I buy from sellers that have lots of feedback and specialize in tubes. I guess I rely on their expertise and honesty. 

There are actually a number of test values that can be measured with a good quality tester. Although I am not sure any have sound quality correlations. 

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The only thing a tube tester can tell you is whether a tube is worn out or nearly so. That you can determine by comparing the manufacturer data to the tube tester result. But the red flag is transconductance.  As tubes wear, transconductance goes down, for any given bias and plate voltage. Consult an RCA tube manual or any of several websites to download the pdf file for the tube under investigation.  These are very old data from manufacturers that no longer exist or no longer make tubes, but the numbers are still valid ways to evaluate tube condition.  If the measured Gm (transconductance) is below the range of values given for a given bias and plate voltage, then the tube is worn or wearing out.  Yet, as Larry said, that tube may still sound "good".  Better test is to replace such a tube with a new one and then listen. If you hear improvement in terms of a livelier sound or better treble and bass response or a sense of better gain character, dump the worn tube or save it for a spare.

The Chinese-made fake ECC81/12AT7 often are labeled as "801S", just to make them look even more desirable.  On the other hand, these fakes often sound fantastic.  I compared about 6 different NOS 12AT7s in the input stage of my Atmasphere amps, many years ago. Other than the Chinese made 12AT7s, all the other tubes were true Euro or US NOS brands such as Amperex, Sylvania, Mullard, GE, etc.  The Chinese tubes sounded by far the best.   I bought them from Billington in the UK, and they were openly sold as Chinese tubes under the label "Billington Gold" (they had gold pins); there was no attempt to mislabel them as TFK or anything like that.  The downside of those Chinese 12AT7s was only that they had a short lifespan compared to true NOS tubes. I used them anyway for a while.

You really cannot tell anything about sound quality from test numbers. In fact, test numbers can be misleading. For example. if you test two tubes of the same type and one has a higher transconductance than the other, it is assumed that tube will have more gain. But only if the plate resistances are the same, otherwise the tube with the higher gain will be the one with the highest combined Gm and plate resistance.

To further complicate things, if two different tube tester are used... do they both have the same voltage parameters for plate, screen and grid to cathode? Your best bet is to purchase tubes from reliable sellers such as Vintage Tube Services or Upscale audio where they test their stock against each other with the same testing equipment. This way you can compare tube readings among the tubes they sell. Getting different readings from different sellers -- especially eBay -- is futile IMO.