Starting with the one before this, my articles contain a segment of cricket-related prose/poetry/anecdote for readers to savour. This is from English Cricket by Neville Cardus.

At half-past two Rhodes and Robinson went out to inspect the wicket, I with them. Rhodes pressed a finger into the soft turf, saying, "Emmott, it'll be sticky at four o'clock," said Rhodes. Emmott simply replied, "Aye, Wilfrid," which was not good enough for me, not good enough for Robinson. So I, in my report, made him reply to Rhodes's "It'll be sticky at four o'clock," "No. Wilfrid, half-past." I put words into his mouth that God intended him to utter.

This article is in two parts. In the first part, I describe a major change made to the PQI (Pitch Quality Index) based on the reader comments to my Golden Willow 25 articles ranking the top Test batting performances of all time. This is the first of the major changes I will be making to the Red Cherry 25/Golden Willow 25 base parameters. In the second part, I analyse the toughest innings played on difficult pitches and with little support, using the newly developed dual-PQIs.

Suggestion: Use PQI for innings 1/2 and 3/4 separately as against the match-PQI
This is a very good suggestion because the average runs per wicket (RpW) for innings three and four is around 10-12% lower than those for the first two innings. Run-scoring is more difficult as the match goes on. For instance, the numbers for the recently completed Perth Test are 43.6 for the match, 52.3 for the first two innings and 34.2 for the last two, indicating that batting became quite difficult as the match went on. Thus, Usman Khawaja's 72 will be valued more. Many thanks to Giri, who suggested this.

On the other hand, in the last Trent Bridge Test, the match PQI is 47.2, indicating a middling pitch slightly tilted towards the bowlers. However, the PQI for the first two innings is 36.7 and for three and four, 56.9, indicating that the pitch improved as the match went on. Thus, Virat Kohli's 97 in the first innings will get a boost in the rankings. However, it is clear that this needs a deep study, especially in regard to matches that lasted three innings.

The PQI is dependent on five values: The Expected runs from batsmen, Actual runs scored, Expected wickets from bowlers, Actual wickets taken, and the Actual RpW. To start with, the following ratios are determined.

- Actual runs to Expected runs
- Actual wickets to Expected wickets
- Actual RpW to Average RpW

Each of these ratios has a range of 0 to 3.0. The sum of the ratios, which is the base PQI, has a range of 0.0 to 9.0. However, since the median of this distribution is around 3.0, a mapping is done to work out a 100-based PQI. The raw PQI of 3.0 maps on to 50.0. A low PQI indicates a bowler-friendly pitch and a high PQI indicates a batting pitch.

The PQI can never be for one innings since that would mean it will be for one team and violate the basic requirement that the PQI should be across teams. As such, the following tweaks will be introduced.

1. For all innings wins, the single match-PQI will be used. This makes sense. The third innings is that of the losing team. It has no leg to stand on, to be considered independently.

2. For three-innings draws, the single match-PQI will be used. In most cases, the third innings is a match-saving one for the team behind and cannot be considered independently.

3. Extending it further, a fourth-innings score of, say, 15 for no loss cannot be considered as a full innings. Hence I will consider the fourth innings worthy of independent consideration only if a minimum of 15 overs are bowled (an hour of bowling) or a wicket taken. This means a fourth-innings score of 150 for 0 will get separate consideration.

This leaves us with the following Tests for which the two-PQI values will be introduced.

- All run wins
- All wicket wins in which the fourth innings has progressed reasonably
- All four-innings draws in which the fourth innings has progressed reasonably

The PQI will be determined considering the two concerned innings together. In other words, for all the 22 batsmen innings of innings one and two, or innings three and four.

I have given below a table of Tests that saw the widest swings, in either direction, of the PQI values. Note how the match-PQI values settle into the benign 50-60 range, leading to all the batting and bowling performances being evaluated on this middle value. This is decidedly unfavourable to the players who performed well in the difficult half of the matches. In the revised scenario, the performances will be evaluated based on the appropriate PQI values.

Just a few examples: Kumar Sangakkara's 185 and Azhar Ali's 302 not out are overvalued and Kieran Powell's 110 and Faf du Plessis' 81 are undervalued in these Tests. The recent New Zealand-Sri Lanka Boxing Day Test is in the table. The match PQI was a nice 50.1, which would have given the innings of Tom Latham and Henry Nicholls a lot of credit. The real situation was quite different. The PQI for the first two innings was 22.2 and for the latter two innings, a huge 76.3. So these hundreds, praiseworthy as they were, need to be valued accordingly, at lower levels.

The last Test is the Centenary Test. It is clear that both Rod Marsh's and Derek Randall's hundreds came when the pitch had become much better to bat on. The PQI changed dramatically from 21.4 to a comfortable 64.6. The highest score by 22 batsmen in the first two innings was 40, and the second pair of innings had two hundreds and innings of 40-plus. These are the Tests in which the dual-PQI concept really comes into its own.

A few interesting facts

- Since the PQI-1/2 and PQI-3/4 have a wider range of values than the PQI-Match, the highs and lows are different. The Australia-England Test in Melbourne in 1932 has the lowest PQI-1/2 value of 14.0 (team scores: 36 and 153). The India-New Zealand Test in Delhi in 1955 has the highest PQI-1/2 value of 93.4 (450 for 2 and 531 for 7). The England-Australia Test at The Oval in 1896 has the lowest PQI-3/4 value of 13.0 (84 and 44). The Ashes Test at the Gabba in 2010 has the highest PQI-3/4 value of 92.5 (517 for 1 and 107 for 1).

- Out of the 2238 Tests played so far, 869 have only single PQI values. 1469 Tests have differing PQI-1/2 and PQI-3/4 values.

- Of those 1469 PQI-1/2 values, 89 are lower than 30.0 (really tough, bowler-friendly conditions); and 89 have PQI values greater than 70 (bowlers' graveyards). The median is 48.8.

- Of the PQI-3/4 values for those 1469 matches, 141 are lower than 30.0, and 112 have PQI values greater than 70. The median is 46.2.

- In the 869 Tests that have a single PQI value, it is below 30.0 for only 20. As many as 107 have values greater than 70. The median is 54.3. This is a clear indication that the match-PQI values are evened out across the four innings and the high number of sub-30 PQI-1/2 and PQI-3/4 values will allow us to evaluate the performances in a much better manner.

Tough innings
The second part of this article has been inspired by a stray comment from a South African reader. This analysis will look at those tough-as-nails innings that were played in the face of great adversity. Support at the other end was virtually non-existent and the batting conditions were near-diabolical. For once, I will not bother about the result. In fact, most of the innings selected have been in losing causes.

Fortunately, I do not need any special analysis to unearth these innings. I already have the widely accepted High Scoring Index (HSI), which is a composite of the support received and the share of the team score. It has been perfected and has been accepted by almost all readers as an excellent measure.

The timing of this analysis is apt, since I have just completed a major revamp of how PQI is determined. Now I can evaluate the batsmen's innings appropriately. For the pitch conditions, I will use the revised PQI values. Going back to last year's Christchurch Test, Tim Southee's invaluable first-innings 68 was played when conditions were really tough (team scores of 178 and 104: PQI of 22.2). Hence this innings will be evaluated based on this value rather than the match-PQI value of 50.1.

I have decided that I will only look at innings of 50 or more runs. The reasons for this are obvious. We cannot glean much from Victor Trumper scoring 18 out of 36. It is too short an innings to have any serious impact. Once I do this, further selection will be done setting 30 as the upper limit for PQI; a value of 30 indicates a difficult pitch giving plenty of assistance to the bowlers.

A few examples from 2018 are the Mirpur Test at the beginning of the year (222 and 110; PQI 27.3), the latter two innings in the Edgbaston Test in August (180 and 162; 28.1) and the Galle Test between Sri Lanka and South Africa (190 and 73; 23.8). Also, Australia's innings win in 1932 (36, 153 and 45; 14.0). I know that this will rule out over 90% of innings pairs (one and two, and three and four). And that is how it should be. It is also essential that the HSI is greater than 1.0 (a fifty out of 100 with the next highest innings being 25 will have a HSI of 1.0).

Giving these two measures equal importance, I have worked out a composite index, called Tough Runs Index (TRI), with a maximum of 100 points. Fifty points are allotted for a PQI of 10.0 (a muddy road anywhere - possible scores of 50 and 50) and 50 are allotted for a HSI of 5.0 (only one HSI in history is greater than 5.0, and that innings does not qualify). Only 46 innings qualified since these are tough parameters. Then I selected the top innings from the table ordered on this composite index. The results are fascinating - a collection of heroic, tough and unforgettable but often forgotten innings.

At the MCG in 1904, England managed a good score of 315. Australia were dismissed for 122. In England's second innings, Johnny Tyldesley scored 62, with a supporting innings of 10 by Albert Relf, and took England to the relative safety of 103. Since Australia scored only 111 in their second innings, the PQI-3/4 was a very low 19.1 (indicating a particularly nasty country road in the outback). Tyldesley's high HSI (4.05) and the very low PQI took his TRI to 75.2 and the top of the table.

An important point is to be noted here. Tyldesley scored 62 in the tough half of the match. Other than his innings, the other 21 produced a total of 152 runs. So, in a way, Tyldesley's innings increased the PQI from a miserable 15 to 19-plus.

West Indies found the going tough in Madras in 1979. Their 228 was answered by India's 255. In their second innings, West Indies could only manage 151. Larry Gomes scored 91 of those runs with the best supporting innings being only 15, by Alvin Greenidge. Since India reached their target of 125 for seven wickets, the PQI-3/4 was 25.7. Gomes, with a very good HSI (3.73), is comfortably placed in second position with 61.1 TRI points.

During the Brisbane Test of the 1949-50 Ashes series, after Australia reached 228, the pitch turned into a gluepot following heavy rain. England declared at 68 for 7, Australia, in turn, declared at 32 for 7, and by close of play, England had dived to 30 for 6 on a day when 20 wickets went down for 130 runs. Len Hutton came in to bat on the fourth day and scored a magnificent 62 out of the 92 added. It was a strategic move to have Hutton bat so low down. Since the next highest score was 17 by Freddie Brown, Hutton's HSI was 1.97, but the PQI for the second half was a measly 15.8 - a powerful testimony to how atrocious the conditions were for batting. Hutton's TRI is 60.0. Hutton's innings is among the all-time great bad-wicket innings ever played.

When we talk of the West Indies tour of England 1976, the Oval Test dominates: the 14-wicket haul of Michael Holding and 291 by Viv Richards. We tend to forget what happened a fortnight earlier, at Old Trafford. On a difficult pitch, Gordon Greenidge played one of the greatest first-day innings ever. The numbers tell us what happened: West Indies scored 211, out of which Greenidge scored 134 and the next highest score was 32 by Collis King. England were dismissed for 71, making the PQI for the first half of the match a low 24.0. The other 21 innings produced 148 runs. Greenidge's innings had an excellent TRI of 54.7. West Indies went on to win by a huge margin.

Against England in Hamilton in 2008, New Zealand scored 470 and took a lead of over 120 runs. They then set England nearly 300 to win. England collapsed to a meagre 110. Out of this small total, Ian Bell scored an unbeaten 54, with a supporting innings of 13 by Alastair Cook. The PQI for the second pair of innings was 24.9 - a steep fall from 63.4 in the first innings. Bell's TRI was nearly 48.

It was a tough Test for Pakistan against South Africa at Newlands on their 2007 tour. On the opening day, Pakistan could only score 157 against a potent South African attack. Mohammad Yousuf came in at 47 for 3 and scored 83 out of the 110 runs added after his arrival. This is a non-typical innings in this collection: a run-a-ball masterpiece. The PQI was a low 26.2, since South Africa could only take a 26-run lead. Yousuf secured 47.4 TRI points.

In Port-of-Spain on the 1995 tour of the West Indies, Australia collapsed to 128 against a powerful West Indian attack led by Curtly Ambrose. Out of this small total, Steve Waugh scored 63 despite coming in at 14 for 3. David Boon's 18 was the nearest support innings. The PQI for the first two innings was 24.0 since West Indies could only secure an eight-run lead. For his sterling effort, Waugh secured 44.9 TRI points.

In St Lucia on the 2016 tour to the West Indies, India scored 353 in their first innings and after securing a lead of 128, set West Indies 346 to win. They could only manage a paltry total of 108. Darren Bravo scored 59 of these. The 12 by Marlon Samuels was the nearest support. The PQI for the third and fourth innings was 29.7. For his fighting innings, Bravo secured 44.9 TRI points.

An experimental English team was dismissed for 92 in Cape Town on the 1899 tour of South Africa. The home team took a useful 85-run lead. Out of 177, Jimmy Sinclair scored 106, with a supporting innings of 26 by Frederick Kuys. The PQI was 28.6. That England scored 330 and dismissed South Africa for 35 is irrelevant as far as this innings was concerned. For this magnificent innings, Sinclair received 44.3 TRI points.

England won the first Test of the Bodyline series comfortably. The second Test was at MCG. Australia scored 228 and England replied with 169. In their second innings, Australia scored 191, which was enough to give them their only win in the series. Out of this, Don Bradman scored a superlative 103, Victor Richardson's 32 being the only support worth mentioning. Bradman only scored two more fifties in the series. The PQI was 25.1 since England could only reach 139. Bradman's magnificent lone-ranger innings secured 43.1 TRI points.

In the Ashes series of 1907-08 at the MCG, Australia scored 214 and this was enough to get them a huge first-innings lead of over 100 runs. England's 105 contained a masterclass of 57 by Jack Hobbs. The only support was George Gunn's innings of 13. The PQI was 29.0, more due to Australia's decent first innings. The TRI value for Hobbs' innings was 43.0.

In Dacca in 1959, when West Indies visited, Pakistan could only score 145 in their first innings. Wallis Mathias came in at 22 for 4 and scored 64 top-quality runs. Shujauddin supported him with 26. The PQI was a measly 21.3 since West Indies collapsed for 76. The TRI for Mathias' innings was 42.8.

A few other important innings are not covered in this table. Matthew Hayden's superlative 119 (38.0 TRI pts) in Sharjah, Rahul Dravid's brave 76 (38.8) against New Zealand in 2002, and Sachin Tendulkar's fighting 51 (38.1) in the same series are in the top-25. Michael Clarke's magnificent 151 (36.6) at Newlands is just outside. Michael Slater's 123 (35.1), in the SCG Test of the 1999 Ashes is in the top 40.

All these innings have many admirable qualities: the lack of support, the contribution to the team score, and the extreme bowler-centric conditions. The result is immaterial. We can only think about "Casabianca" and the burning deck. Contrast these runs to meaningless hundreds on flat pitches.

A few words on a historic win by India down under. Notwithstanding the favourable toss results and the absence of two top Australian players, this is an excellent, well-deserved win. While it was expected that the Indian batsmen would be far ahead of their Australian counterparts, no one expected that the Indian bowlers would also be thus. It was as much a bowling win as a batting win.

Readers will know that I always give equal importance to the bowlers. My next article will be a similar one on tough spells of bowlers. The dynamics of the two analyses are delightfully different. I will also cover the major changes I have made to the Bowling Quality parameter, with respect to the Golden Willow 25/Red Cherry 25.

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